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LIBRARY OF CONGRESS, 


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UNITED STATES OF AMERICA. 















































































. 







The Nonpareil System 


—OF— 


Hand 


Railing. 


ORIGINAL IN CONCEPTION, 

SIMPLE IN THEORY, and 



UNIVERSAL IN ITS APPLICATION. 



John V. H . Secor, 

J M 


Practical Stair Builder. 



1889. 









/ 



Entered, according to act of Congress in the year 1888, by John V. H. Secor, 
in the office of the Librarian of Congress at Washington, D. C. 





INTRODUCTION. 


I HAVE been a practical Stair Builder for many years, and in my 
daily work have come in contact with many of the different 
systems of Handrailing that are in use. I have thus been 
able to learn in what repute they are held by the trade. For the 
most part, they are considered abstruse and difficult of application 
in practical work. A number of my fellow-workmen, being some¬ 
what familiar with the peculiarities of my own practice, have long 
desired me to publish a book by which it should be given to the 
world. They have urged, in support of this request, that the utility 
of my method would secure for such a volume preference over others 
which have preceded it. I have, however, hesitated to act upon this 
suggestion. I began teaching the system as far back as 1863, and 
in the interval have given instruction to a large number. Mean- 
• while a book has been published, in which parts of my system are 
explained. This circumstance would seem to be calculated to leave 
in the minds of some persons the impression that the author of the 
book referred to originated the lines he has used, but which, in fact, 
are mine. In support of this, I have the signatures of a number of 
those whom I have taught—a precaution taken against just such an 
infringement of my rights as has occurred. It would now seem that 
the time has arrived when, in justice to myself, and as the best service 
I can render to my fellow-workmen, I should publish my system. 
This I do in the following pages, choosing as a name for the work 
the “Nonpareil.” 



IV 


INTRODUCTION. 


In presenting a system of Handrailing, original in its general 
features, it is necessary to give careful attention to each problem in 
course. By this means alone can the principles underlying the system 
be fully defined. Accordingly, my aim in the pages following has 
been to lead the learner, step by step, from the simplest problems to 
those of the most complex character, so defining principles and illus¬ 
trating methods as to make him competent for any emergency that 
may arise in practical work. 

A leading feature of this work is the mode of ascertaining the 
length of mould. It is a simple method, and the resulting dimen¬ 
sion is called the major length. The lengths of tangents are then 
applied to produce the mould. The minor length is found in the 
same manner, forming the parallelogram, giving a point from which 
to set up the various hights and differences in hights and to find the 
width of the moulds on the minor length. 

The system of bevels illustrated herein, I believe, will at once 
commend itself to the student as being simple as well as universal 
in application. The use of the trammel in drawing the curves of the 
moulds, and also the method of finding the length of the trammel rod, 
it is thought will be found easy, and something to be appreciated 
by those who desire to acquire efficiency in the art of describing 
elliptical curves with the trammel. An original mode of locating the 
minor axis is another point. The “sub-normal,” or parallel to normal 
or minor axis, is also original, and will be found to be an easy mode 
of locating the minor axis on moulds of other than right angle bases. 
As will be observed, it does away with many lines which have here¬ 
tofore been used, but which are, in reality, superfluous. 

In order to make the system of easy comprehension, I have first 
introduced the simplest problems, using the fewest lines for illustration. 
Following are problems with a gradually increasing number of lines, 
presented for the purpose of fully illustrating the principles of the 
work. Following these in turn are problems again decreasing in 



INTRODUCTION. 


v 


the number of lines to the close of the book, finishing with those 
with as few lines as it is possible to use in Handrailing. 

The work is also arranged in such a manner as is believed will 
make the study of the various problems comparatively easy. The 
diagrams and letter press are arranged in a way to render reference 
from one to the other quite convenient. 

This book is not intended to teach the art of stair building, 
but rather to enable anyone in the line of joinery to draw the moulds 
for any kind of stairs, however unscientifically they may have been 
constructed. It is no uncommon occurrence to be called upon to 
put the rail on stairs that have been built by some one that has 
never given a thought to the rail—how it is to be put on. This 
work, it is believed, will be of positive help in all such cases. 

After the necessary pages devoted to Glossary and Simple 
Geometrical Problems, there are presented eighteen problems which, 
in effect, constitute the key to the system. In these problems the 
portion of cylinder to be covered by a rail, and the tangents in 
elevation, are given. Following these problems there are a number 
of others, selected on account of their frequent occurrence in practical 
work, and which still further illustrate the principles under discussion. 

With these introductory remarks, I present to my fellow-workmen 
the system on its merits, believing that if they will give it careful 
attention, the most satisfactory results will follow. 

JOHN V. H. SECOR. 


New York , February , 1889. 







































































































' 








Table of Contents. 


I 


Introduction, - 
Glossary, - 

Simple Geometrical Problems and Definitions, - 

The Principles of Drawing Hand Railings, 

Illustrative Problems, - 

General Practice, - 

Splayed Work, - 

Bevels and Mitres, - 

Determining Width of Mould in Special Cases, - 

Sliding the Mould, - 

The Nonpareil System in its Briefest Form, 



GLOSSARY. 


PITCH. —Rake or hypothenuse, as the inclined side of the pitch board. 

NORMAL or LEVEL. —As the minor axis is always level, there 
being no twist or cant on this line. 

SUB-NORMAL. —Parallel to normal, used in this work to direct 
the minor axis for acute and obtuse angle, ground plan having 
two pitches. 

MAJOR LENGTH. —Greatest length from which the mould is 
drawn. 

MINOR LENGTH. —Crossing major, but not necessarily the short¬ 
est ; in a full easement over a ground plan of more than a 
quarter, it is the longest; a term peculiar to this work. 

MAJOR AXIS. —Longest or transverse. 

MINOR AXIS. —Shortest or conjugate, never changing its length 
from that of the ground plan, and crossing the major axis at 
right angles. 

STRAIGHT-WOOD. —That portion which is added to a mould or 
rail outside of the curve and parallel to the tangent. 

OVER-WOOD. —The portion of material to remove in the opera¬ 
tion of squaring up the twist to form a handrail. 

TWIST. —A curved piece of wood used in handrailing. It is formed 
by applying bevels at the ends, giving the plumb line through 
the centre to form a twist. 

BUTT JOINT. —A square joint to connect handrails to each other, 
as the centre but joint. 



GLOSSARY. 


IX 


PARALLELOGRAM. —A figure having its opposite sides parallel 
and of the same length, 

RIGHT ANGLE. —Base or ground plan; a quarter cylinder is a 
right angle base and from this all angles are measured or 
reckoned. 

OBTUSE ANGLE.— Base or ground plan. In this the curve of 
the cylinder is less than a quarter circle, so the angle is formed 
obtuse on the line of the tangents and is said to cover less than 
a quarter circle, while the angle is greater than a right angle. 

ACUTE ANGLE. —Base or ground plan. In this the curve of the 
cylinder passes the right angle or quarter, so that the angle on 
the line of tangents is acute and is said to be more than a 
quarter circle, while the angle is less than a quarter or right 
angle. Thus, if we draw a circle and inclose it within tan¬ 
gents, there will be four right angles ; if acute, then three ; if 
obtuse, then an indefinite number. 

EASEMENT. —A full cylindrical easement is formed by a level and 
a pitch tangent. The starting and landing piece would be a 
full easement. 

HALF EASEMENT. —An easement formed when the tangents are 
of different pitches. 

RAMP. —A piece of rail connecting two pitches, forming a curve up 
and down, while its sides remain straight. 

OVER-EASEMENT. —The piece connecting the flight or pitch with 
the level rails. The top of a platform stairs would be an over- 
easement, while the starting from the level or from a Newel 
would be a simple easement. 








I 


SIMPLE GEOMETRICAL PROBLEMS 

« 

AND DEFINITIONS. 


The inclination of two lines meeting one 
another, or the opening between them, is called 
an angle. Fig. i is called an equilateral tri¬ 
angle, all sides being of one length, forming three 
acute angles —a, b and c. This figure is als 
called a trigon. 

Fig. 2 is a right angle triangle, having 
three unequal sides, forming one right angle, d, 



Fig. i.—Equilateral 
Triangle. 



This 


figure 


is 


Fig. 2.—Right Angle 
Triangle. 


and two acute angles, as at e and f, 
also called a scalene triangle. 

Fig. 3 is an obtuse angle triangle, having two 
of its sides equal, forming an obtuse angle, g ; also 
two acute angles, h and j. 

Lines may be straight or curved, having length with¬ 
out breadth or thickness. A line may be composed 
of points or dots, or it may be a continuous line, 
yet its ends will be points. A straight line is the shortest distance 

between any two points. A vertical line -7J 

is a plumb line, and at right angles to 
the horizon. A horizontal line is a level 
line and parallel to the horizon. A line 
may be perpendicular to a given line 
without being in a vertical position. 

A point has position without magnitude and 
is represented by a dot, or by two lines meeting at 
an angle, or crossing each other, a, b and c, Fig. 
i, and a, Fig. 6, are points. 

To draw a perpendicular upon the end 
of a line. Let a b in Fig. 4 be the given line 
and b the given end. With radius less than a b, 
Fig. 4 .—To erect a Per- set one foot of the compasses in b and then in a; 

PENDICULAR UPON THE •, Li. • , ,•_. ^ npi VL „ 

end of a given line, strike short curves intersecting at d. I hen with d 



Fig. 3.—Obtuse Angle Triangle. 










2 


SIMPLE GEOMETRICAL PROBLEMS AND DEFINITIONS. 



as centre and a b as radius, describe a portion of a circle, a b c, indefin¬ 
itely. Then from a through d draw a line, cutting the circle at c. 
Connect c b. Then b c will be the perpendicular sought. 

m To erect a perpendicular at the end of a given 
line. Let h j in Fig. 5 be the given line. Set one foot 
of the compasses in j, and with j h as a radius, describe 
the arcs h l equal to h j. Draw a line through h l in¬ 
definitely; make l m equal h l. Connect m j, which will 
be the perpendicular sought. 

Fig. 5.—Another method 
OF ERECTING A PERPENDICULAR 
AT THE END OF A GIVEN LINE. 

To find the length or stretch¬ 
out of a semi-circle. Let a, Fig. 

6, be the centre of the semi-circle 
bcd. Set one foot of the compas¬ 
ses in d, extending the other to b, 
and describe the arc b e. Then 
setting one foot in b in the same 
manner, describe d e. Draw a line 
from e through d and another from 
e through b indefinitely. Draw n m 
parallel to b d, touching the semi¬ 
circle at c. Then n m will be the stfetch-out of the semi-circle, or so 
nearly equal to it as to answer all practical uses. We may find the 

stretch-out of any portion of a circle 
by the same means. For example, 
take d o upon the semi-circle as the 
curve, the stretch-out of which we 
wish to find. Draw a line from e 
through o, cutting the line n m at g. 
Then g m is the length of the curve from o to d and g n equals ocb. 

To find the stretch-out of a cylinder in another way. Let 
h j, Fig. 7, be the radius, or half of the diameter 
and equal to a c of Fig. 6. Make h d three times 
the length of h j. Connect d j. Then d j will be 
the stretch-out or a line in length equal to n m of 
Fig. 6. 

To bisect a line is to divide it in two 
equal parts. 



Fig. 6.- 


-FlNDING THE STRETCH-OUT OF A 
Semi-circle. 



Fig. 7.—Another method of doing 
the same. 


L 


W 


XM 


In Fig. 


8 the line w x is bisected FlG - 8 _E T,n™ g * c,veN 














SIMPLE GEOMETRICAL PROBLEMS AND DEFINITIONS. 


3 



Fig. 9.—Describing a Circle 

TO BASS THROUGH ANY THREE GIV¬ 
EN POINTS. 


by l m at right angles to it. Let w x be the 
given line to bisect. Take any distance 
greater than x d for a radius and w and x for 
centres, and describe the arcs intersecting at 
l and m. Connect l m, which will then 
bisect w x. 

To find the radius to describe the 
circle which will pass through any three given 
points not in a straight line, as x y z in Fig. 9, 
proceed as follows : From x and y as centres 
with any convenient radius longer than one- 
half the distance from x to y, describe short arcs intersecting at h and k. 
Draw h k. In the same manner describe arcs from y and z as centres, 
intersecting at l and j. Draw l j, and prolong it until it meets h k, 
thus establishing the point b ; then with b as a centre and b x as a radius, 
describe the circle which will cut x, y, and z. 

A circle may be defined as 
a round figure, bounded by a 
single line in every point 
equally distant from a point 
which is called the centre. In 
Fig. 10, c is the centre of the 
circle. 

The circumference or 
boundary line of a circle is 
supposed to be divided into 
360 equal parts called degrees; 
each degree, in turn, is divided 
into 60 equal parts called min¬ 
utes, and each minute into 60 
equal parts called seconds. A 
degree is the 360th part of the circumference of any circle, without 
regard to the size of the circle. The radius of a circle is a line drawn 
from the centre to the circumference, as c 0, Fig 10. The diameter of 
a circle is a line drawn through the centre terminating at the circum¬ 
ference, as 0 x 180, Fig 10. A semi-circle is a half circle, as 0 x 90 
x 180, Fig 10. 

A chord is a line crossing a circle, cutting off a portion, as a b, Fig. 
10. A segment is the portion cutoff by a chord, as a b, 270. The rise 



Fig. 10.—Showing circumference, diameter, 

RADIUS, CHORD, SEGMENT AND TANGENT. 













4 


SIMPLE GEOMETRICAL PROBLEMS AND DEFINITIONS. 


of a segment is the distance from the chord to the circumference; for 
example, d, 270, Fig 10, is the rise of the segment just described. 

A tangent to a circle is a line drawn outside of the circle touch¬ 
ing the curve at a single point, as the line g h which touches the circle 
at the point 40 in Fig. 10. 

To find the tangent to a circle, proceed as follows : From c as 
centre, draw c r, passing through the circle at the point to which the 
tangent is to be drawn. Take any distance, as f 40 for radius, and with 
one foot of the compasses in the point 40, set off the points e and f, 
then with e and f as centres, strike short arcs intersecting, as shown at 
G and h. Draw g h. Then g h will be the tangent sought. It will 
also be a line at right angles to the radius produced, touching the circle 
at the required point. 



Fig. 11.—Drawing the Ellipse. 


We will next consider 
the ellipse, which may be 
drawn by a string as shown 
in Fig. 11. The ellipse 
is described from points 
located on lines called the 
“ axes ” of the ellipse, and 
which cross each other at 
right angles. The line 
running through the fig¬ 
ure lengthwise, as a b, Fig. 

11, is known as the major 
or transverse axis. The 
shorter one or c d of Fig. 11, is known as the minor or conjugate axis. 
To draw the ellipse a, d, b, c, with string and pencil, we proceed as 
follows : Take the length of one-half the major axis, as e b, and with 
one foot of the compasses in d strike a short arc, cutting the major axis 
in f, also in g. Then the points f and g, called foci, are the places in 
which pins are to be placed ; also drive a pin at d. Stretch a string 
around the points f d and g, remove the pin at d and put in place of it 
a pencil, and keeping the string drawn tight against the pins, f and g, 
describe the curve a d b c. 

To find the tangents to a given ellipse determine, first, at which 
point a joint is to be made: for example, h, Fig n. Then draw the 
dotted lines g h and f 1. With one foot of the compasses in h, with any 
convenient radius, make points on the lines intersecting at h, as indi- 












SIMPLE GEOMETRICAL PROBLEMS AND DEFINITIONS. 


5 


cated by i, 2, 3 and 4. With same radius, using 1, 2, 4 and 3 as 
centres, describe short arcs intersecting at 5 and 6. Connect 5 and 6. 
Then the line 5 6 will be the tangent sought. Make the joint at right 
angles to the tangents passing through h. 

Fig. 11. To find the joint first, and afterward to draw the 
tangent from the joints, proceed as follows : Let k be the point at 
which the joint is to be made. From the foci g and f draw lines inter¬ 
secting at k, as shown. Then with k as centre, with any convenient radius, 
establish the points l, n, t and x. With l and x as centres, describe 
short arcs, intersecting at 7, and from n and t as centres, describe short 
arcs, intersecting at 8. Draw the line 7 8, which will be the joint 
sought. In this case the tangent may be drawn by erecting a line per¬ 
pendicular to 7 8, and cutting it in the point k. 

A spiral may be defined as 
a single continuous curved line 
between two points, as from a to 
s, Fig. 13. It can be construct¬ 
ed by segments of circles to an¬ 
swer all practical purposes, as 
laid down in Figs. 12 and 13. 

A scroll may be defined as form¬ 
ed by two spiral lines, as the 
outside and inside of a hand-rail. 

The outside line is called the 
convex spiral, and the inside line 
the concave spiral. The two 
lines completing the scroll term¬ 
inate at a point called the ball, 
or the eye. The reader will ob¬ 
serve that the two scrolls illustrated in Figs. 12 and 13 are drawn in 
the same general manner. Fig. 12 may be described as a close scroll, 
and Fig. 13 as an open scroll. 

To construct the scroll shown in Fig. 13, proceed as follows: 
Determine the size or spread you wish to have the scroll between out¬ 
side lines, as a b. Draw b c at right angles to a b. Connect a c by 
the diagonal line, which will pass through the eye. Divide a b into two 
equal parts. With one foot of the compasses in a and b, with any con¬ 
venient radius more than half of a b, describe arcs intersecting at d. 
Perpendicular to a b draw the line d f. Divide b c in a similar manner 


















6 


SIMPLE GEOMETRICAL PROBLEMS AND DEFINITIONS. 


and draw e f parallel to a b, 
cutting the diagonal line at the 
point f. From h as centre, with 
h f as radius, describe the arc f 
j. Draw j k l at right angles to 
a b. With k as centre, and k j 
as radius, describe the semi¬ 
circle j l, which will pass through 
the eye. Draw the diagonal line 
from j, passing through the eye. 
From l also draw a diagonal 
line through the eye, as shown, 
terminating in the point r on the 
line f e. From l drop the per¬ 
pendicular l c. Proceed in like 
manner from the points r, s, v, 
&c. With this done, to draw the spiral, take j as centre and j a as 
radius, describing the arc a i. Then from l as centre, with l i as 
radius, describe the arc i c. Then from p as centre, with p c as radius, 
describe the arc c 2 and proceed in like manner with r s, &c., as 
centres. For the inside or dotted line, proceed as follows : From j as 
centre, with j 6 as radius, describe the arc 6 7 ; and from l as centre, with 
l 7 as radius, describe the arc 7 8, and so on. 

It is to be observed that the line b c in both Figs. 12 and 13 govern 
the proportions of the scroll. As this line is lengthened the scroll will 
be made open ; if it is shortened the scroll will be closed ; compare 
Figs. 12 and 13. Matters of this kind are always left to the taste of the 
stair builder. 

In Fig. 14 is illustrated 
the method of describing 
an ellipse with a trammel 
and rod. The trammel 
consists of two pieces of 
wood or metal crossing 
each other at right angles, 
and having a groove run¬ 
ning the entire length of 
each piece. The rod is 
made to carry a pencil at 


B 



Fig. 14.—Ellipsb drawn with trammel and rod. 


A 































SIMPLE GEOMETRICAL PROBLEMS AND DEFINITIONS. 


7 


one end and is provided with two movable heads, with pins on the 
under side to fit in the grooves so as to slide easily. The trommel is 
secured in place on lines representing the major and minor axes of the 
ellipse, as a c and d b. e is the pencil. In setting the trammel make e 
f equal to b j, and e h equal to a j. Then move the rod, letting the 
pins slide in the grooves : the result will be an ellipse, as shown. 

Fig. 15 illustrates a method 
of finding the radius by which 
to describe a segment to a given 
chord and rise. Let a Bbe the 
given chord, and c d the rise. 

Bisect a b, establishing the 
points d, and erect the perpen¬ 
dicular d c. Extend c d indefi¬ 
nitely in the direction of 0. 

Bisect a c, establishing a point 
j. From j as centre, with any 
convenient radius, longer than 
j a, strike short arcs intersect¬ 
ing in the points k and h, as shown. Draw k h and produce it until it 
intersects c d produced in the point o ; then 0 will be the centre and 0 c 
the radius of the arc a c b. In the second case, illustrated in Fig. 15, 
let a b be the given chord and e d the given rise. Proceed in the same 
manner as shown, which will establish the centre at l. 



In Fig. 16 is shown an 
original method for find¬ 
ing the distance to set a 
gauge to eight-square any 
sized timber. Let a b c 
d be a square of 12 inches. 
From a as centre, des¬ 
cribe the quadrant d b, 
also the quadrant g o, 
having the radius a g, 
which must be 3^ in¬ 
ches in length. Lay 
down the size of timber to octagon as at 4 e. Extend 4 e, touching the 
quadrant at p. Then draw from p toward a,' cutting the inner circle at 



Fig. 16.—Original method for setting the gauge to 

EIGHT-SQUARE TIMBER 

































8 


THE PRINCIPLES OF DRAWING HAND-RAILINGS. 


gauge 


will be a b, 
inches, and so on. 


r. Draw r s parallel to a g. Then the required distance to set the 
gauge is from a to s, as shown by d t at the right. 

In like manner any piece may be octagoned from i to 24 inches. 
Example : Using a scale of 3 inches to the foot, for 12 inches the 

or 3^4 inches; for 24 inches it will be 7 
The size of material to octagon is on the 
line d to c, and the gauging is found 
at a 0. 

Another way to find the gauging 
is shown in Fig. 17. Let a b be 24 
inches long, and b c 7 inches. Let a d 
be the size of timber, touching the 
diagonal line at e. Then e d will be 



G 


Fig. 17.—Another method of , 

Eight-Squaring. the gauging, as shown at E F D. 

Fig. 18. To draw an Octagon from 
a given side. Let a b be the given side. 

Draw the square a b c d. With a as a cen¬ 
tre, describe the quadrant a b d. Connect 
a c, and where the diagonal line touches the 
circle at E'will be one angle of the octagon. 

Extend a d indefinitely and make a l equal 
to a b. Parallel to a d, make e f equal to 
F. a. Draw e k and f j at right angles to e f. 

With the compasses set equal to a e, space 
f g, h j, and k l. Connect them, complet- 


H 



K 


ing the octagon. 


Fig. 18.— To draw an Octagon 
from a given side. 


THE PRINCIPLES OK DRAWING 
HAND-RAILINGS- 


Problems showing the mode of Ascertaining the Length 
of Mould, the Stretch-out of Tangents, Forming the Mould 
and Drawing the Curve Line by the use of the Trammel. 

Fig. 19. From a quarter circle, ground plan, to draw three 
moulds. Let a b c d be the ground plan ; a b and b c the tangents ; d 
the centre from which the curve is drawn. Extend d a and c b indefin¬ 
itely. From b as centre, describe the dotted line c g. Extend the line 



























THE PRINCIPLES OF DRAWING HAND-RAILINGS. 


9 


g g parallell to c b, which is 
the stretch-out of tangents. 
From f to g draw the pitch. 
To find the major length of 
the mould, take a as the centre 
and a c as the radius and des¬ 
cribe the dotted line c h. Con¬ 
nect h and e, which is the 
length required from which to 
draw the mould. 

In drawing the mould we 
will draw tangents and centre 
curved line only, giving the 
trammel rod for the curve. 
The mould completed will be 
shownYurther on. As we grad¬ 
ually lead the way, the reader 
will be able the better to com¬ 
prehend. 

To draw outline of 
mould from plan and ele¬ 
vation. Draw i 3, Fig. 20, 
equal to e h ; 12 equal 

to e f, and 2 3 equal to 

f g. At right angles to 1 2 
and 2 3, make 4, which gives 
the parallelogram, and is also 
a quarter of an ellipse. Extend 
1 4 for the major axis, and 
extend 3 4, which will be the 
minor axis. To find the 
length of trammel rod to draw 
the curve. Make 5 6 equal to 
3 4, and 5 7 equal to 1 4. As 
the points 6 7 are moved along 
the lines of axes, the point 5 
will describe the curved line as 
required, and when set up in 
position will be plumb over 
ground plan a b c d, i 5 3 
covering a 0 c. 



Figs. 19, 20, 21 and 22.—Showing method of ascer¬ 
taining LENGTH OF MOULD AND STRETCH-OUT 
OF TANGENTS. 

























IO 


THE PRINCIPLES OF DRAWING PIAND-RAIL1NGS. 


The minor axis is a level line, and is the point at which the bevels 
change their inclinations. As Fig. 20 requires but one bevel, it is 
applied at the point 1, as will be explained hereafter. 

Fig. 21. Elevation for a mould having tangents of equal 
pitches to cover the ground plan. From the base line, a h g, set up 
the hight to e. The major length is found in the same manner as in 
Fig. 19. The pitch line, e g, is the two tangents, x being the centre. 
Draw x o parallel to g h a, which is the minor length and the minor axis. 


Fig. 23. To draw the 
outline of mould, make 

1 3 equal to e h; Fig. 21. 

With the compasses at 3, 
make the points 2 4 equal 
to e x. From 1 bisect at 

2 4 ; connect these points 
as shown, forming the 
parallelogram 1234. 

Connect 2 4, extended, 
which will be the minor 
axis. Draw 6 7 parallel 
to 1 3, touching at 4, which will be the major axis. Make 4 5 equal 
d o of ground plan, then 5 will be the centre of curve line and 1 2 and 2 3 
the tangents. 

To find the length of 
trammel-rod, make 3 8 
equal to 4 5. Extend 
this line to the minor axis, 
which will be the length 
of rod. 8 9 are the pins 
and 3 the pencil, which 
will describe the curve. 

Fig. 22. Elevation 

Fig. 24.-T0 DRAW OUTLINE OF MOULD. for a mou fo having 

tangents of unequal pitches. Let a e be the hight, e x one 
pitch and x g the other. Connect e h for the major length. Draw 
the minor x o parallel to the base line a h g. 

Fig. 24. To draw the outline of mould, make 1 3 equal to e h, 
Fig. 22, 1 4 and 2 3 equal e x, and 1 2 and 3 4 equal x g. Connect 
these points and form the parallelogram as shown at 1 234. 



2 










ILLUSTRATIVE PROBLEMS. 


11 


To find the position of the axis. Let i 6 equal e o. Con¬ 
nect 6 4, which will be the minor axis. At right angles to 6 4 draw 7 8, 
touching at 4, which will be the major axis. 

To find the centre of curve on the major axis. Let 4 9 
equal do of ground plan, then 9 will be the point required. 

To find length of trammel-rod. Let 3 5 equal 4 9, crossing 
the major axis. Extend this line to the minor axis, then 3 5 o is the 
length required, o and 5 being the pins, and 3 the pencil, which will 
describe the curve. 


ILLUSTRATIVE PROBLEMS. 



The following eighteen problems 
embrace all the different forms of 
cylindrical hand-railing, drawn to a 
scale of inch to the foot, and hav¬ 
ing straight wood on both ends, which, 
in practice, is sometimes required. 

They are intended to fully illustrate 
the principles, and will be referred to, 
occasionally, in some of the other 
problems. 

Fig. 25. To draw the stretch¬ 
out and pitch of tangents. Let a b 

cd be the ground plan, with abc the 
tangents, e the centre of curve line, 
a to f the full hight. From b, as 
centre, describe the dotted line c g. 

Connect g f, which will be the tangents. 

To find the major length. 

With a as centre, describe the dotted 
line c h and connect h f, which will be the length required. Extend 
c b, touching the tangents at j. Draw j k parallel to base line, as the 
minor length, and it will stand plumb over the base, bxd, 

Fig. 26. To draw outlines. Let 1 3 equal f h. From points 
1 3 draw tangents equal to f j and j. g. Connect 2 4 extended, which 



C D 

Fig. 25.—Showing quarter-cylinder, 

GROUND PLAN AND ELEVATION. 














12 


ILLUSTRATIVE PROBLEMS. 


is the minor axis. At right angles from 2 4 draw the major axis. For 
the centre of mould let 4 5 equal d e. From 5, lay off half the width 
of rail on each side as 6 7. Draw the dotted line 5 1 and parallel to it 
draw 6 o and 5 o touching the cord line at 00. The outer dotted lines 
are the width of rail. Draw the straight wood parallel to the tangent 
1 2 and the other end of mould will be the same width. 


To find the 
bevel to square 
up the rail. Draw 
the dotted line at 
right angles from 
2 3, touching at 
4. Take this 
length and place 
it from b to l, 
when the angle at 
l will be the bevel 
for both ends. 

To find the 
length of tram¬ 
mel-rod for outside curve. Let 8 9 equal 6 4 and extend to 10. 
For inside, let 0 x equal 7 4 and extend to 12, which will be the lengths 
required. 



Fig. 26.—Showing outline of mould, mode of finding bevel and 

TRAMMEL ROD FOR CURVES; ALSO, WIDTH OF MOULD AT ENDS. 


A very easy way 
to make the curve 
is to take a small 
rod and mark the 
point upon the 
edge to corres¬ 
pond to those 
already described. 
Operation : Let 
9 10 be placed at 

Fig. 27.—Showing mould completed. 

different points 

along the axis, and at the end of rod at 8. Make a mark, using as 
many points as desirable, and then trace the curve through the points 
thus formed, or use a flexible strip holding it in such a manner as to touch 
all the points, and mark the curve along its face, completing the mould. 

Fig. 27 is the mould completed and will not need explanation. 















ILLUSTRATIVE PROBLEMS. 




Fig. 28 shows a quarter cylinder, ground plan and elevation, with 
tangents of unequal pitch, requiring two bevels for squaring up the rail. 
The position of trammel is also changed, as will be seen at Fig. 29. 

To find the major length, tangents and bevels. Proceed as 
per directions, Fig 26. 

Fig. 30 shows the mould complete, also points from which to get 
the bevels. Let abcd, Fig. 28, be the ground plan, with e as the 
centre of curve. From a to f set up the hight. From f to j draw 
pitch, and from j to g will be the other pitch. From f to h will be the 
major length. Draw j k parallel to base line, a b g. 

Fig. 29. To draw out-lines, let 

1 3 equal f h. With the compasses 
take the length of tangent f j, and 
place one foot in 1. Describe the arc 
at 2, then with one foot in 3 describe 
4. Next take length j g, and from 1 
describe the arc bisecting at 4. Then 
from 3, bisecting at 2, connect 1 2 
and 2 3 for the tangents. Connect 
1 4 and 3 4, which will be the chord 
line, 4 being the point or centre of 
axis. To find position of axis let 3 5 
equal f k, and draw the minor axis 
through 4 and 5. Draw major axis 
at a right angle to 4 5 ; touching at 4. 

To find centre of mould on 
minor axis, let 4 7 equal d e. Find 
width ofrail from 7 as the centre to 6 8. 

To find the width of mould at the ends, 
draw the parallel dotted lines 678, 
touching the chord line 1 4- Draw the straight wood parallel to tan¬ 
gents from the points 0 0, and repeat the operation at 3. 

To find length of trammel rod for outside curve, let 9 12 
equal 4 6; extend touching at 13, which will be the length required. 

For inside let o 14 equal 4 8 ; extend touching at 15, which will 
be the length required. 

To find bevels. At right angles to 1 2, Fig. 30, from 5 draw the 
dotted line touching 4. From 2 3 at 6 in the same manner draw 6 4. 
Upon the elevation let b m equal 4 5 ; then the angle M will be the bevel 



Fig. 28.—Quarter cylinder, ground plan 

AND ELEVATION, AND TANGENTS OF 
UNEQUAL PITCH. 















H 


ILLUSTRATIVE PROBLEMS. 


for the wide end of mould. For the other end, let b l equal 4 6, then 
the angle at l will be the bevel required for the other end. 

Ground plan and elevation of a quarter cylinder, having a 
single pitch line, the other being a level line. Fig. 32 is a mould 

such as is used in starting or in landing, as in connection with 


winders. Fig. 33 is a mould similar to 


Fig. 


straight wood is on the opposite end 
landing for a straight flight of stairs. 

Fig. 31. Let 
a b c d be the 
ground plan of 
a quarter-cyl¬ 
inder, e centre 
of curve as des¬ 
cribed from d. 

From a set up 
the hight re¬ 
quired to F. 

Connect f d, 
which will be 
the pitch, and 
also the length 
of tangent to 
draw the 
mould. The 
angle at f gives 
the bevel to be 
applied at the 
wide end of 
mould. No 
bevel is requir¬ 


32, except that the 
This is used in starting and 



Fig. 30 

Figs. 29 and 30. —Showing outline and mould complete. 


ed for the opposite end, as the rail is there taken square through the centre. 

Fig. 32. Let 1 2 equal f d , and 2 3 equal b c. Square from or 
at right angles to 1 2 and complete parallelogram at 4. Let 5 6 be the 
width of rail. Draw dotted lines, as indicated, parallel to 1 3, giving 
width of mould as required. 

To find length of trammel rod to describe the curve. 

This being a quarter ellipse, 1 4 is the minor and 3 4 the major axis. 
For outside length let 9 o equal 4 8, and o 12 equal 6 4 ; then 9 12 are 
the pins and o the pencil to describe the curve. 








ILLUSTRATIVE PROBLEMS. 


IS 


Find the 
inside in the 
same man¬ 
ner. Let 13 
14 equal 4 7, 
and 14 15 

equal 4 5. 
This applies to 
Fig. 33 as well. 
All numbers in 
Fig. 33 corre¬ 
spond with 
those of Fig. 
32 - 



Fig. 34. An obtuse base, or less than a quarter, having 

tangents of one pitch. Let abce be the ground plan, with d forming 
parallelogram, and e the centre from which the curve is described. 
Find the stretch-out of tangents. From b as a centre, and c as a radius, 
describe the curve c g.- From a to f set up full hight, and connect eg, 
giving the length of tangents required. At right angles from a b draw 
b j, bisecting the tangents at j. From a as a centre, and c a as the 
radius, draw the curve c h, and connect f h, for the major length of 
mould. Draw j k parallel to b a, giving the normal point and minor 

axis, as will be seen at Fig. 35. 

Fig- 35. To draw the mould. Let 
1 3 equal f h, 12 and 1 4 equal f j, and 
3 2 and 3 4 equal j g. Let 2 5 equal b e, 
then 5 will be the centre of the axis. 
Connect 5 1 and 5 3 for the chord lines. 
For the centre of rail upon the normal and 
minor axis, let 5 6 equal e l, and 7 8 the 
width of rail. Draw the parallel dotted 
lines to find the width of mould at the 
ends, as described at Fig. 29. Make the 
straight wood parallel to the tangents. 
Find the length of trammel rod for out¬ 
side, and let 9 12 equal 8 5. Extend to 



Fig. r». —Mould used in starting and . • . •• i , 1 • , 

3 landing straight. minor axis at 13, giving length required. 


























i6 


ILLUSTRATIVE PROBLEMS. 


For inside, let o 14 equal 5 7, extend 
to minor axis at 15, giving length required, 
and this setting will describe the curves. 

For bevel take length of dotted line 4 x, and 
make d m on ground plan equal; then the 
angle at m will be the bevel required for 
both ends. 

Fig. 36. An obtuse base or ground 
plan having tangents of unequal pitch, 
requiring two bevels. Let a b c d 

. be the base, with e as the centre, from 
which the curve is described. To find the 
stretch-out of tangents from b as centre, 
describe the curve c g. To find pitch and 
length of tangents : From a to f set up 
the full hight required. From f apply the 
pitch, touching the vertical line at k, then from k to f will be the long 
tangent. Connect k g, giving the short tangent. 

To find the major length of mould with a as centre : Describe 
the curve c h, and connect h f, giving length required. From k draw 
k j parallel to a b as the point for the sub-normal. The normal proper 
is on the minor axis, as will be explained further on. 

To find the 
minor length of 
mould : Extend 
the tangent g k to 
to m, as indicated 
by the dotted line; 
then from m to f is 
what is called the 
difference in hight. 
For example : we 
will suppose the 
hight of the long 

tangent to be 12 inches, and that of the short tangent 8 inches ; then 
the difference between the two is 4 inches, which is found in this prob¬ 
lem at m f. On the minor base line e b, erect perpendiculars from the 
points e and d, make d n equal to m f, and connect b n, extended to 
o; then from b to o is the minor length. For the width of mould on 


8 




Fig. 34—Obtuse base with Tan¬ 
gents of one Pitch. 













ILLUSTRATIVE PROBLEMS. 


17 


Erect the 



Fig. 36. —Obtuse base with unequal pitch 

TANGENTS. 


this line, lay down width of rail from l as the centre to p b. 
perpendiculars touching at s r. 

Fig. 37. To draw the 
mould. Let 1 3 equal f h ; let 
1 2 and 3 4 equal f k, and 1 4 and 

3 2 equal g k, completing the 
parallelogram. Connect 2 4 ex¬ 
tended, making 4 5 equal to no ; 
then 2 5 is the minor length, 5 
being the corresponding point 
over e of the ground plan. For the 
chord line, connect 5 1 and 5 3. 

To find the position of the 
axis. Make 1 6 equal j h, and 
connect 6 4, giving the sub¬ 
normal line. Draw 5 7 parallel 
to 4 6, then 5 7 is the minor axis 
and the normal proper. Draw the major axis at right angles to 7 5. 

For width of mould on minor length. Let 289 equal bsr, 
Draw parallel dotted lines to give the width at the ends described in the 
previous problems. 

For the bevels, square from the tangents as indicated by the dotted 
lines, touching at 4 10 and 4 12. Let d u on the ground plan equal 

4 12, then the angle at u is the bevel for the wide end of mould. Let 
d T equal 4 10, then the angle at t is the bevel for the narrow end. 

To find the 
length of tram¬ 
mel rod for in¬ 
side curve. Let 
13 14 equal 5 o, 
and extend to minor 
axis at 15, giving 
length required. 
The outside will be 
found in the same 
manner as already 

described in the previous problems. 

Fig, 38. An obtuse base or ground plan having one pitch 
and one level tangent, requiring two bevels. The normal line of 














i8 


ILLUSTRATIVE PROBLEMS. 



Fig. 38.— Obtuse base with one pitch and 

ONE LEVEL TANGENT. 


this kind of mould will be outside of the parallelogram; consequently the 
bevels will both be applied from one side. 

Fig. 38. Let a b c d be the base 
or ground plan, with e as the centre 
from which the curve is described. 
a to f will be the hight, f b the 
length and pitch of tangent. Find 
major length of mould, with a as 
centre ; describe the curve c g, and 
connect g f, which gives the length 
required. Let l k be half the width 
of rail, and raise the perpendiculars 
to the points m n. Let d h equal 
a f. Connect b h extended to j, 
which will be the minor length, j being the centre of the axis. 

Fig. 39 - Let 1 3 equal f g, i 2 equal f b, and 3 2 equal b c. Find 
the point 4 in the same manner as at 2, finishing the parallelogram. 
Connect 2 4 extended to 5. Then the points on this line from 2 5 will 
correspond with those from b j. Connect 3 5 extended, giving the 
major axis. From 5 at right angles to 3 5 draw the minor axis. 

2 To find the bevels, 

draw the dotted lines at 
right angles to the tan¬ 
gents 1223, terminating 
at 4, as indicated at 8 and 
9. Let 0 d on ground 
plan equal 4 8, when the 
angle at 0 will be the 
bevel for the narrow end 
of mould. Let p d equal 
4 9, then the angle at p 
will be the bevel for the 



Fig. 39. —Showing manner of drawing mould. 


wide end of mould. For the width of mould on minor length, let 2 6 
equal m n, and 6 7 equal m b. Draw dotted lines, terminating at the 
chord lines 1 5 and 3 5, giving width of mould at the chord line. 

For length of trammel-rod. Let 7 12' equal e k, and extend 
to minor axis at 13, which will be the length required, 7 13 being one- 
half of the major axis 5 14, and will describe the inside curve of mould. 
Find outside length in the same manner. 












ILLUSTRATIVE PROBLEMS. 


19 


Fig. 40. An acute base, or more than a quarter, having 
tangents of equal length, requiring one bevel to be used on 
both ends. Let a b c d be the base of tangents, and e centre from which 
the curve is described. Let a f be the hight. From b as centre, and 
b c as the radiuSj describe c 

the curve, touching base 
line at g. Connect f g, 
giving length and pitch 
of tangents. From b 
draw the perpendicular, 
touching pitch at j. 

For major length of 



mould, with a as centre, 
and a c as radius, de¬ 
scribe the curve, touch¬ 
ing base line at h. Con¬ 
nect f h, giving length 
required. 


D 


C 

Fig. 40.—Acute base, Tangents of equal length. 


Fig. 41. Let 1 3 equal 


f h, 1 2 and 1 4 equal f j, and 3 2 and 3 4 equal j g. Let 2 5 equal 
b e, giving centre of axis. From 5 draw the chord lines through 1 and 
3. Let 5678 equal e k l m; then 6 7 8 is the width of rail on the 
minor axis. Draw through 5 the major axis at right angles to the 


For the bevel, let n d equal 4 3, as indicated by the 


minor axis. 


dotted line, then the angle 
at n will be the bevel. 



Find the width of 
mould at the ends by 
parallel lines already 
described, for tram¬ 
mel-rod. Let 9 11 equal 
6 5, extend to minor axis 
at 12, giving length re¬ 
quired, which will de¬ 
scribe the inside curves. 
Find the outside length 
in the same manner. 


4 

Fig. 41.—Showing manner of drawing mould. 


Fig. 42. An acute base having unequal pitches, requiring 
two bevels. Let a b c d be the base or ground plan, and e 


















20 


ILLUSTRATIVE PROBLEMS. 


centre from which the curve line is described. From a to f set up the 
full hight. From b erect the perpendicular indefinitely. From f apply 
the pitch, cutting the perpendicular at j. From b as centre and bc as 
the radius, describe the curve c g. Connect g j, giving the length of 
short tangent. For major length, take a for a centre and a c as the 
radius, describe the curve c h, and connect h f, giving length required. 

To find the point for sub-normal, from j draw the line j k 

parallel to a b, cutting the major length at k. 

For minor length of mould, extend the tangent g j, as indicated 
by the dotted line, cutting the perpendicular at y. Then erect the 
perpendiculars from d e o p r. Let d m equal f y, and connect m b, 


v r 



Fig. 42. —Acute base with unequal pitches. Fig. 43. —Manner of drawing mould. 

which gives the minor length of mould as required. Then stu will be 
the width of mould as applied at Fig. 43. 

Fig. 43- Let 1 3 equal f h, 1 2 and 3 4 equal v j, and 3 2 and 
1 4 equal j g, completing the parallelogram. Make all points on the 
minor length from 2 4, equal all points from b m. To find the sub¬ 
normal, let 1 9 equal h k, and connect 9 4, giving sub-normal. From 
5 draw the minor axis parallel to 4 9, and at right angles draw the major 
axis. The minor axis is the normal point of mould as explained in 
Fi & s - 37 ” 39 - Again the points 4 and 5 are reversed from those in Figs. 37- 
39, etc. 4 in each case forms the parallelogram, and 5 corresponds to the 














ILLUSTRATIVE PROBLEMS. 


21 



ground plan at e, and from which the curves are described. Then 5 
will be plumb over the point e, when in position, and must be the 
centre of axis. The chord lines are drawn from 5 to 1 and 5 to 3. For 
the width of mould at ends, proceed in the manner already described, 

no further explanation 
being required. 

For bevels in like man¬ 
ner as described. dx equals 
4 o ; then the angle at x is 
the bevel for the long tan¬ 
gent. Let v d equal 4 12; 
then the angle at v is the 
bevel for the short tangent. 

For length of tram¬ 
mel-rod. Let 13 14 equal 
eo, extend to minor 
axis at 15, giving length 
required. Find outside 

Fig. 44. —Acute base with one pitch and one . . . 

level tangent. length in the same manner. 


Fig. 44. An acute 
base or ground plan 
having one pitch and 
one level tangent, requir¬ 
ing two bevels. Let a b 
c d be the ground plan of 
tangents, with e as the centre 
from which the curve is de¬ 
scribed. From a to f, set 
up the hight, connect f b, 
giving pitch and length of 
tangent. For major length, 
with a as centre and a c as 
radius, describe the curve c g 

d t. „ ~ • • .1 Fig. 45.—Manner of drawing mould. 

connect f g, giving the 

length required to draw the mould. From d e k l m erect perpendiculars. 
Let d h equal a f, connect h b. h b will be the minor length and nop 
the width of mould required. 














22 


ILLUSTRATIVE PROBLEMS. 


Fig- 45 - Let i 3 equal f g ; i 2 and '3 4 equal fb; 32 and 1 4 
equal b c, completing the .parallelogram. Connect 2 4, and make 
4 5 equal h j. 5 is the centre of axis, and the lines 5 1 and 5 3 will 


be the chord lines. 5 3 is also the 
major axis. Draw minor axis at 
right angles. For width of mould, 
let 6 78 equal p 0 n. For the 
width at the ends, find in the usual 
way by parallel lines. For trammel- 
rod, let 6 9 equal e k, extend to 
minor axis at 0, giving length 
required. 

For bevels, let e r equal 5 12; 
the angle at r will then be the bevel 
for the long tangent. Let e s equal 
3 5, when the angle at s will be the 
bevel for the short tangent. 



Fig. 46.—Quarter ellipse with tangents 

OF EQUAL PITCH. 


Having completed the nine problems as including all those of a 
true circle, we will now take those employing the Elliptical form, follow¬ 
ing in the same order and treating them in the same manner, viz : 
That the base or ground plan must have the short sides or tangents made 
equal to the long sides, because in hand-railing it is necessary to cross 
the curve as nearly square as possible. By connecting b d, P'ig. 46, we 

cross the curve line at o. 

Fig. 46 shows a quar¬ 
ter Ellipse with tangents 
of equal pitch. Let abfe 
be the portion to draw the 
mould, c d being added. 
From a to g set up the 
hight. From b as centre 
and b c as radius, describe 
the curve, touching at j. 
Connect g j, givingthe pitch and length of tangent. Extend c f b, cut¬ 
ting the tangent elevation at h. From b describe the curve f k, erect a 
perpendicular from k, cutting the tangent elevation at l. For major 
length of mould, with a as centre, and a c as radius, describe the 
curve c m. Connect g m, giving length required. It will require 


2 



Fig. 47.—Manner of drawing the mould. 














ILLUSTRATIVE PROBLEMS. 




additional hight in proportion as the short tangent has been lengthened ; 
this will be found at k l as the hight added. For the width of mould 
from o as the centre make n p equal to the width required. 

Fig. 47. To draw the 
mould. Let 1 3 equal g m, 

1 2 and 1 4 equal g h, 32 and 
3 4 equal h j; then complete 
the parallelogram. To make 
the joint on short tangent, let 

2 8 equal h l and make the 
joint square from tangent 2 3. 
For the width of mould in 
centre, let 4 5 6 7 equal d n 
o p. From 567 draw the 
dotted parallel lines in the 
usual manner. From 9 o 
draw back parallel to the tan¬ 
gent, cutting the joint at 14 15. 
For outside curve use the flex¬ 
ible strip, letting it touch the 
found in the same manner 



■ o 


is 


UNEQUAL PITCH. 

points 15 7 13. The inside curve 
from 14 5 12, completing the mould. 

For bevel, let d r equal 4 16, the angle at r being the bevel for 
both ends. 

Fig. 48. A quarter ellipse with tangents of unequal pitch, 
requiring two bevels. Let a b f e be the position to draw the 
mould, with c d added, a to g is the hight. From g apply the pitch 
for the long tangent, 2 

cutting the perpendicu¬ 
lar b h at h. With b as 
a centre, describe the 
curves f k and c j. Con¬ 
nect j h, giving length 
and pitch of short tan¬ 
gent. With a as a 
centre describe the curve 
c m, and connect g m, 
giving the major length of mould. From k draw the perpendicular k l, 
cutting tangent line at l. Extend j h to n, as indicated by the dotted 
















24 


ILLUSTRATIVE PROBLEMS. 





Fig. 50.—Quarter ellipse, single pitch, level 

TANGENT, ONE BEVEL. 


line. For minor length, from d erect the perpendicular d 0 equal to 
n g ; connect o b, giving length required. Let p s be the width of the 
rail. 

Fig. 49. Let 1 3 equal g m ; 

1 2 and 3 4 equal g h ; 32 and 
1 4 equal j h, and 2 8 equal 
h l. At 8 make the joint square 
from the tangent 2 3. Let 4 5 
6 7 equal 0 t u v. From 567 
draw the parallel lines for width 
of mould at the ends, as ex¬ 
plained in Fig. 45. Draw 
curves in like manner through 
13 7 15, also through 14 5 12. 

Find the bevels in manner ex¬ 
plained in all previous problems. 

Fig. 50. A quarter ellipse having a single pitch with 
a level tangent requiring one bevel. Let a b f e be the portion 

to draw the mould, with c d added. From a to g set up the hight. In 
this figure there is no extra hight required, as it takes in the long tan¬ 
gent only. Connect g b, giving pitch and length of tangent required. 

For major length, with a as 
centre and a c as radius, de¬ 
scribe the curve c h. Connect 
g h, giving length required. 
For minor length, let d m 
equal a g, as required. 

Fig. 5 i. Let 1 3 equal g h ; 

1 2 and 3 4 equal g b, and 
complete the parallelogram. 
Connect 2 4 as the minor 
length. Let 4567 equal 

Fig. 51. —Manner of drawing mould. „ T _ 1 „ 1 , 

m n o p, and make 9 1 o equal 

to j k l. hor the width on the wide end at 3, draw the parallel lines in 
the usual manner. For length of short tangent let 2 8 equal b f; draw 
curved lines through the points indicated 12 7 o and 13 5 9. The 

angle formed at g is the bevel to be applied at the wide or level 

end, as at 3. 
















I ILLUSTRATIVE PROBLEMS 


25 



N 


L 





P 

liVI K 1 

J B / 

\ , Ac~ 


Fig. 52. An obtuse base with Elliptical curve, having 
tangents of equal pitch, requiring one bevel. Let a b f e be 

the portion to draw the mould, with c u added. From a to cset up the 
full flight as required for extra length from f c, for length of tangents. 
With b as centre and b c as radius, describe the curves c m. Connect 
g M, giving length and pitch of tangents. With b as a centre describe 
f j, draw the perpendicular j l, cut¬ 
ting the tangent line at l. With a 
as a centre describe c k and connect 
g k, giving major length of mould. 

From b erect the perpendicular b n, 
cutting the tangent line at n. 

Fi g- 53- Let 1'3 equal g k ; 12 
and 1 4 equal g n ; 32 and 3 4 equal 
n m, and complete the parallelogram. 

For joint, let 2 7 equal n l, and 
make joint at 7 square from tangents. 

For width of mould on minor line, 
let 4 6 equal do, and from 6 as the 
centre make 5 2 equal to the width 
required. For the width at end draw 
the parallel lines in the usual manner. 

For the bevel, let d p equal 4 8. The angle at p gives the bevel for 
both ends. 

Fig. 54. Form of base similar to Fig. 52, but with tangents 
of two pitches, requiring two bevels. Let a b f e be the portion 
to draw the mould for with c n added. Find iength and pitch of tan¬ 
gents. From a g as the 


high t, apply the pitch 
from g, cutting the per¬ 
pendicular b h at h, and 
connect h j. Then g h is 
the long tangent and h j 
the short one. With a as 
centre and a c as a radius, 
find the point m ; connect 
g m for major length. For the minor length, from d erect the perpen¬ 
dicular d 0 equal to n g, and connect 0 b, giving the length required. 
From p as the centre of mould and r as half the width, erect the 


c o 

Fig 52.—Obtusf 3Ase with elliptical 
curve Tangents of equal pitch, 

ONE BEVEL. 



F‘c. 53 .—Manner of drawing mould. 



















26 


ILLUSTRATIVE PROBLEMS. 



o 


perpendiculars, cutting the 
minor length o b at s and t. 

Fig- 55 - I .et i 3 equal g m; 
i 2 and 3 4 equal g h ; 14 and 
3 2 equal j h, and complete the 
parallelogram. Let 2 7 equal 
h l, and at 7 make joint square 
from tangent. For centre and 
width of mould, let 4 5 6 2 
equal o s t b. For the width 
at ends, draw the parallel lines 
as indicated and complete the 
mould by drawing the curves. 

For the bevels, let d u equal 
Fig. 54 -Base similar to 5 o, with two bevels. 4 the angle at u being the 

bevel for long tangent. Let d v equal 4 9, when v gives the bevel 
for the short tangent. 

Fig. 56 shows a base similar in form to Figs. 52 and 54, 
but with a simple pitch tangent, and a level tangent requir¬ 
ing two bevels, the same as for a turnout easement. Let a b f e, 
Fig. 56, be the portion to draw the mould, with c n added. From a to 
g set up the full hight and connect g b, giving the length and pitch of 
tangent. For major length, from a as centre and a c as radius, describe 
the curve c h, and connect g h, giving length required. For minor 
length, from d erect the perpendicular d l equal to a g, and connect l b, 
giving length required. 

For width of mould on 
minor length. From 
j as centre of mould, 
and j k as half the 
width, erect the per¬ 
pendiculars, cutting 
the line b l at o and p. 

Fig. 57. Let 1 3 
equal gh; 12 and 3 4 
equal g b ; 14 and 3 2 equal b c, and complete the parallelogram. 
Connect 2 4 as the minor length. For width of mould, let 4 5 6 2 
equal l p 0 b. For width at the ends, draw the parallel lines as 















ILLUSTRATIVE PROBLEMS. 


27 


indicated. For joint on short tangents, let 2 7 equal b f, make joint 
square from tangent, and complete the mould by drawing the curved 
line in the usual manner. For the bevels, let 4 8 equal d r ; the angle 
at r will then be the bevel for the long tangent. Let 4 7 equal d s ; the 
angle at s will then be the bevel for short tangent. 

An acute angle base with an elliptical curve, having tan¬ 
gents of equal pitch, and requiring one bevel to be applied at 
both ends. 

Fig. 58. Let a b f e be the 
portion to draw the mould, with 
c d added. From a to g set up 
the full hight. With radius b c, 
describe c h, connect g h, giving 
length and pitch of tangents. 

With radius a c, describe c k,. 
and connect g k, giving major 
length. From j b erect the per¬ 
pendiculars, cutting the tangents 
at m and l. Draw s t parallel 
to a b, the distance between 
to be equal to the width of mould 
at its normal point, and where 

the bevel d p crosses this line will be the width of the mould at the 
ends p and v. 

Fig- 59 - L et 1 3 equal g k ; 12 and 3 4 equal g l ; 3 2 and 1 4 
equal l h, and complete the parallelogram. Connect 2 4 as the 
minor length. Let 456 9 equal d u 0 r. Let 8 12 equal p v 

as the width at both ends. 
Complete the mould by 
drawing the curved line 
through the . points indi¬ 
cated. For joints on short 
tangent, let 2 7 equal l m, 
and draw square from tan¬ 
gent, giving point required. 
For the bevel, let d p equal 
4 o; the angle at p will 
then be the bevel required 
for both ends of the mould. 



Fig. 56 —Base similar to Figs. 52 and 54, single 

PITCH AND LEVEL TANGENT, TWO BEVELS. 



Fig. 57. —Manner of drawing mould. 












28 


ILLUSTRATIVE PROBLEMS. 


An acute angle base with an elliptical curve, having 
tangents of unequal pitches, requiring two bevels. 

Fig. 60. Let a b f e be the portion to draw the mould, with c d 
added. From a to g, set up the hight. From b erect the perpendicular 

b m, and at g apply the 
pitch, cutting the perpen¬ 
dicular at m for long tan¬ 
gent. From b as centre 
describe the curve f j. 
From j erect the perpen¬ 
dicular, cutting the tangent 
at l. For major length of 
mould, from a as centre, 
describe the curve c k, 
connect g k, giving length 
required. Extend h m to 
n, as indicated bv the dot- 
ted line. For minor length, 
from d erect the perpendic¬ 
ular d o equal to n g; con- 



D 


Fig. 58. —Acute angle base, eli.iptical curve, tangent 

OF EQUAL PITCH. 


nect 0 b, giving length 
required. Let r p s be the width of rail, and erect perpendiculars, cut¬ 
ting the minor length at v u and t. Draw w x parallel to a b and 
equal to r s. 

Fig. 61. Let 1 3 equal 
gk; 12 and 3 4 equal g m. 

Let 1 4 and 3 2 equal h m; 
complete the parallelogram 
and connect 2 4 as the 
minor length. For joint on 
short tangent, let 2 8 equal 
m l. For width on minor 
length, let 4 5 6 7 equal 
o v u t. For bevels, let 
D Z equal 4 I 4 > then the Fig. 59 —Manner of drawing mould. 

angle at z is the bevel for long tangent. Let d y equal 4 15, then the 
angle at y is the bevel for the short tangent. For width of mould at the 
ends, let o 9 equal y w ; 1213 equal x z, and complete the mould by 
drawing the curves through the points indicated. 





















ILLUSTRATIVE PROBLEMS. 


29 



An acute angle base with an elliptic curve, having one 
pitch and ope level tangent, requiring two bevels. 

Fig. 62. Let a b f e be the portion to cover with the mould, c d 
added. From a to g set up the hight; connect g b, giving length and 
pitch of tangent. From a as centre, describe the curve c h and connect 
g h, giving major 
length of mould. 

For minor length, 
from d erect the per¬ 
pendicular d j equal 
to a g and connect 
j b, giving the length 
required. Let k l m 
equal the width of 
rail, erect the perpen¬ 
diculars, cutting the 
minor length at n o 
and p. Draw t u 
parallel to a b and 
equal to k m. 

Let 1 3 equal g h ; 12 and 3 4 equal g b j i 4 and 3 2 equal c b, and 
complete parallelogram. Connect 4 2 as the minor length. Lor the joint 
on short tangent, let 2 8 equal b f. For width on minor length, let 

4567 equal jnop. For 
the bevels, let d s equal 
4 14, then the angle at s is 
the bevel for long tangent. 
Let d r equal 4 15, then 
the angle at r is the bevel 
for short tangent. For 
width of mould at the ends, 
let 9 o equal t r, and 1213 
equal u s. Complete the 
mould by drawing the 
curves through the points 
indicated. 

The plan of a straight flight of stairs, drawn to a scale of y A inch 
to the foot. « The landing-rise is placed at the chord line, the cylinder 


Fig. 60.—Acute angle base, tangents of unequal pitch. 



Fig. 61—Manner of drawing mould. 






















ILLUSTRATIVE PROBLEMS. 


3 ° 


being 8 inches in the clear. The centre of the rail is drawn ]/ 2 inch on, 
as indicated by the dotted line. The risers are 7^ inches and the treads 
10 inches, which is the size of the pitchboard. 

Fig. 64 is the plan of stairs. 

a, centre of cylinder; b b, centre of 
rail and tangent line ; c, joint 
through centre of the cylinder. 
a 1 and a 2 are the risers; d, centre 
of the short baluster, at which 
point the bottom line of the rail 
must rest. 



Fig. 62.—Acute angle base tangents, single 

BITCH AND LEVEL. 


Fig. 65 shows the piece from 
which the rail is to be made 
placed in position, and with the 
bevel applied at e. 

Fig. 66 shows the level 
mould drawn over the ground plan of cylinder from the centre at a. 
b x is the straight wood, which may be varied as to length, according to 
circumstances, and will be made from 3^2 or 4 inch plank. 

Fig. 67 is the piece as it will be when squared up. f g shows the 
centre line or tangent, from which the mould is to be drawn. 

Draw the chord line b b, 
extended indefinitely. Draw 
from c as the centre of mould 
parallel to b b indefinitely. 

From the line f c apply the 
pitch at f, giving the inclina¬ 
tion of tangent f g through 
centre of plank, which must be 
3/"2 or 4 inches thick. 

Fig. 68. To draw the 
mould, let 1 2 equal f g, 

Fig. 67, and 2 3 equal a c, 

Fig. 64. From the parallelo¬ 
grams, with 4 as the centre of the axis, let 6 8 equal the width of 
rail, 314 inches. Let 5 7 equal h j, Fig. 67. 

Fig. 69. To find the length of rods to strike the ellipsis. 
For the outside curve, let 9 o equal 4 8, Fig. 68 ; and let 9 12 equal 



13 


Fig. 63.—Manner of drawing mould. 















GENERAL PRACTICE. 


3i 


4 7, Fig. 68. For inside, let 13 14 equal 4 6, Fig. 68 ; and let 13 15 
equal 4 5, Fig. 68. The rods as applied are shown at Fig. 68, and for 
fuller explanations refer to Figs. 31-33. 


GENERAL PRACTICE. 



Fig. 70 shows the level piece of rail 
squared up, having an easement, a b, 
thickness of plank; c n, thickness of rail ; 

r ’ Fig. 67. 

e, the floor line ; c, the under side 01 rail, 

4 inches from the floor, and at f, 3^ 
inches, giving an ease¬ 
ment of p2 inch, complet¬ 
ing the turn. A plan of 
stairs starting with a curve, 
commonly called a turn- A 1 
out or offset, 
showing how 
to calculate 
the flight of the 
newel. 


Fig. 66 . 

Fig. 64. _Plan of stairs. Fig. 66.—Level mould on ground plan. 

Fig. 65. —Piece from which rail is made. F ig. 67. —Section of rail squared. 

Fig. 69.— Scale. 


In this case the hight is made to agree with the rail, and will give 
a full easement in the rail. Scale, inch to the foot. 

Fig. 71. I ,et a b c d be the ground plan, with e as the point from 
which the curve is described; the lines 1234 represent the rise lines, 
and the broken lines the ends of tread. The dotted lines are the centre 
of rail, and the points as described on this line are the centre of balusters. 
In locating the position of the rail, the short baluster is used as the 
point of contact for the bottom of the rail, 00 being the width of rail. 










































32 


GENERAL PRACTICE. 



Fig. 68.—Manner of drawing mould. 


The centre of the newel is located on the face of the first rise. The 
mitre is drawn first to determine the end of the rail or point of the 
mitre. The line h h, crossing at the intersection of the rail and cap, is 
called the cheek line, and is used to make the mitre in the cap as the 
guide to square into the centre. 

For length and pitch of tangent. From the base line a b 
apply the pitch-board, and from the point touching at b extend the 
pitch, cutting the perpendicular line a f at f; then b f is the length of 

tangent and a f the hight. b c is the 
length of level tangent. For major 
length of mould, with a as centre, and 
a c as radius, draw c g, and connect f 
g, giving length required. For minor 
length, from d erect the perpendicular 
d j, equal to a f ; connect b j, extended 
to k, cutting the perpendicular erected 
from e. From o o erect the perpen¬ 
diculars, cutting the line b j k at l l 

• 

Having made all the lines from which to draw the mould, draw it on 
the board or paper, which is to be cut out for use, thereby saving time. 

Fig. 72. Let i 3 equal f g; 12 and 3 4 equal f b; 14 and 
3 2 equal b c. Connect these points, completing the parallelogram. 
Connect 2 4, extend to 5, equal to b k. From 5 as the centre, draw 
lines crossing at 1 and 3, which will be the chord line ; 3 and 5 is also 
the major axis. Having fully explained the axis in the previous prob¬ 
lems, hereafter we will give the points through which to draw the curves, 
using the flexible strip as the most convenient to accomplish the 
desired end. 

For the bevels, set the compasses equal to 
n n, place one foot in 4, describe the segment 6 , 
draw a line parallel to 1 3, touching the segment 
at 6. For the bevel at 1, set the compasses in 4, 
extend to the tangent line 1 2, as indicated by the 
dotted line; describe it until it touches the line at 
7, draw back to 4, then the angle at 7 is the bevel. For the other end, 
extend the compasses to 8 , as indicated by the dotted line, describe the 
curve, touching at 9, and draw back to 4; then the angle at 9 is the 
bevel for the end at 3. For the width of mould at the ends, let 6 o 
equal the width of the rail, draw the dotted line as indicated parallel to 



Fig. 70.—Showing level 

PIECE SQUARED, 





























GENERAL PRACTICE. 


33 


7 9, cutting the bevel lines at 12 13. For the end at 1, make it equal 
to 7 13; and for 3, make it equal to 9 12. For centre, let 4 x x equal 
j ll, Then through the points thus made draw the curve, completing 
the mould. 

Fig. 71. For the hight of Newel, let m be the under side 
of the rail, over the short baluster s, on the third step from m. Drop a 
perpendicular to the floor equal to three risers, as shown at p. Add 

2 feet 2 inches as the hight of the rail over the short baluster to the 
hight p s, less half the thickness of the rail, which will be the hight of 
the newql from the floor to the underside of cap, making in this case 

3 feet 6 inches. 



E 

Fig. 71. —Plan of stairs with turnout curve. 


Fig. 72. For the mitre line or cheek, draw 14 14 parallel 
to the joint 3 5, and the same distance from 3 as it is on the ground 
plan from c. In squaring the rail, the mitre is left until ready to put 
on the cap, leaving the under side at the end flat and square from the 
joint, in order to set the cap on a line, after which the surplus wood can 
be removed, and the easement finished. The sections of rail 15 16 
show the bevels as applied. The stock of the bevel is on the top 
or face side. 

Fig. 73 shows the ground plan of stairs starting with winders, 
and having a newel located same as in Pig. 71. 1 he pitch line is on 

the same inclination as the straight rail, no romp being required. I he 























34 


GENERAL PRACTICE. 


plan of this is more than a quarter, and forms an acute angle on 
the line of tangents. Let a b c d be the ground plan of tangents, e the 
centre from which the curve is described. From b apply the pitch-board 

extending to the perpen¬ 
dicular at f. Find the 
major length in the same 
manner as described in 
other problems as at g f, 
and produce the minor 
length in the usual man¬ 
ner from the hfght a f; 
apply at d j, b j being the 
minor length. P'or the 
hight of newel, set off half 
the thickness of rail from 
pitch line fb at m, over 
short baluster at s. From 
m drop the perpendicular 
Fig. 72.—Manner of drawing mould. through the Centre of 

short baluster, equal to five risers, to the floor at p. Add 2 feet 2 inches 
to the hight p s, less half the thickness of rail, which will be the hight 
from floor to under side of the cap, which in this case will be 4 feet. 

Fig. 74. To draw the 
mould in the usual man¬ 
ner. Let 1 3 equal f g ; 12 
and 3 4 equal fb; 14 and 
3 2 equal b c. Complete the 
parallelogram. Connect 2 4, 
and let 4 5 equal j k. From 
5 find width of mould, also 
the bevels from the same point. 

Let 5 6 equal e a, and let 6 o 
equal the width of rail, the 
width at the ends being shown 
at 8 13 and 9 12. Sections 15 
16 show the bevels as applied. 

It will be seen that they do 
not apply in the same manner 

as in Fig. 72, but Cross each Fig * 73—'Ground plan, starting with winders. 

other, because the angle is acute. 





























GENERAL PRACTICE. 


35 


Easements of tins kind—that is, having one pitch and one bevel, 
have the bevels applied in the following manner : A right angle has one 
bevel applied on the level end, the opposite end being square. The 
obtuse base, as in Fig. 71, has the bevels applied from one side, as in 
ldg. 74. The bevels 
cross each other, because 
the minor axis is within 
the parallelogram, while 
the axis of the obtuse 
angle is outside. This 
will be so easily under¬ 
stood as to need no 
further explanation. 

The four following 
figures exhibit a plan of 
8 inch cylinder stairs, 
with the landing rise 2^ 
inches in the cylinder: 
rise 7^4 inches, tread 9 Fig. 74-—Manner of drawing mould. 

inches. This will require two pitches in the lower piece, and is what 
is known as a half easement, while the top or landing piece is a full ease¬ 
ment. Rail, 2^4 inches thick, by 3)4 wide. Scale 1 ]/ 2 inch to the foot. 

Fig. 75. Pro¬ 
ceed to find the 
length and pitch 
of tangents, and 
the major length 
of the mould. 
The dotted cuiwe 
line is the centre 
of the rail, with 
the points show¬ 
ing the location 
of the balusters. 
Let a b c d be 
the quarter to 
draw the mould, 
with a half ease¬ 
ment, and c e f d 



Fig. 75.— Plan of 8-inch cylinder stairs. 





















































36 


GENERAL PRACTICE. 


the portion for a full easement. Let x x be the risers as drawn on the 
plan. Extend a b indefinitely for the base line. Extend the tangent 
lines on plan e c b indefinitely, also f d a. Stretch out the tangents by 
placing one foot of the compasses at b extended to c, and describe the 
curve c g, also e h, from the same point. Erect perpendiculars from 

2 g and h indefinitely. 

Draw the elevation as 
shown by the double 
lines at any convenient 
distance from the base 
line a b. From o o as 
the location of the short 
balusters on plan, let j j 
Fig. 76.—Manner of drawing mould. represent the same on 

elevation, and connect j j for the bottom line of the rail. From j j set 
off half the thickness of the rail; extend this line until it cuts the per¬ 
pendicular e c b. On the perpendicular h, set up 4 inches to the under 
side of the rail; add half the thickness of the rail as at l ; connect k l, 



fpy/////' 


1 


Fig. 77.—Showing flexible strip cf rattan. 

giving the length and pitch of the short tangents, which will correspond 
to the plan be. n k m will be the tangents for the mould at Fig. 76. 

From n draw n p parallel to a g. The full hight contained in the 
mould will be found at p m. Let p r equal a c on the plan Fig. 75, and 
connect m r, which will be the major 

1 

length of the mould, Fig. 76. From k 
draw k s parallel to b g, cutting the 
major length at s. 

To draw the mould, let 1 3, Fig. 76, 
equal r m, 12 and 3 4 equal km, 23 
and 1 4 equal n k, and complete the 
parallelogram. Let 1 5 equal r s and 
connect 4 5 extended indefinitely, which 
will be the minor axis and normal line. 

Draw the major axis at right angles to 
4 5 through 4. For the centre of the 
rail let 4 6 equal d a. From 6 set off half the width of rail as 7 8. 
Find the width at the end by the parallel dotted lines. For the bevels, 



3 


WM 

W//////M 

D D 


'//M/////A 

. .. 


Fig. 78.—Manner of drawing mould. 































\ 


GENERAL PRACTICE. 


with the compasses, take the length 4 9 and place it on the plan f t ; 
the angle at t will then be the bevel for section a a as applied. Let f v 
equal 4 o ; the angle at v will be the bevel at section b b as applied. 
Add whatever straight wood is required from 3 parallel to the tangent. 
Complete the mould by drawing the curve through the points as given 



by the dotted lines, using a flexible strip or the trammel rod, as may be 
most convenient. 

The best flexible strip for all moulds is a piece of rattan planed down 
at one end to y% of an inch and the edge squared, leaving the other 
end about y 2 inch thick, as shown at Fig. 77. This will naturally form 
an elliptic curve as it is bent, using the thin end at the short tangent. 

Fig. 78. Let 1 2 equal f e (Fig. 75) and 2 3 equal l m. At right 
angles to tangent from 1 and 3 draw the lines crossing at 4, giving the 












































38 


GENERAL PRACTICE. 


'A A 


B B 


axis, 3 4 being the minor axis and normal line, and i 3 the major axis. 
At 3 lay off the width of rail, draw the parallel lines to find width on 
the opposite ends, add the straight wood and complete the mould by 
drawing the curve. The bevel is found at l, the pitch being the bevel 
and applied at section c c. At section d d the rail is taken square 
through the centre. The stock of the bevel is shown as applied from 
the top or face of the stuff in all cases. 

A plan of two top portions of stair landing with four 
and five winders —Figs. 79 having five, and 80, four winders. The 
straight portion is the s..me for both, using the same ramp-scale, 3 ^ inch 

2 to the foot, 7^2 inches rise, 

10 inches tread, 12 inches 
cylinder. Rail 2*^ x 3^ 
inches. The dotted line 
shows centre of rail on 
which the balusters are 
spaced off, as shown at Fig. 
79, and represented by 
dots. The centre of rail will be regulated by the size of balusters ; in 
this case the centre of rail is 1 inch on. 

Draw the elevation by setting off from the 
chord line a b three spaces equal to c n as d c 
e f. This line will also be the floor line, from 
which drop down three risers to the point g; 
then draw the elevation of the steps outside of 
the cylinder, as h j k. At j k will be the bot¬ 
tom of the rail, as it cuts the treads at the centre 
of the short balusters. From f set up 5 ]/ 2 inches 
to the centre of the rail on the landing, as at 
l, and connect l h, giving the length of the 





__—- 9 ; "^7 




12\\8 

/ \\ 


4 



Fig. 81.—The mould. 



D D 


tangents for Figs. 81 and 82. 



touching the perpendicular at p. 


iiiiiiiilimiiil 

Fig.'82.—The mould. 

From f drop down one rise to m, which 
will be the floor line for plan, 
Fig. 80. From m set up 5^ 
inches to the centre of the rail 
at n, connect n o for the tan¬ 
gents of Figs. 83 and 84. At 
the intersection of the tangents 
with the chord at z, square off 
From p to s is the hight contained in 































general practice. 


39 


Fig. 8 1, and will cover the plan, Fig. 79. a t c and p r will also cover 
the same portion at Fig. 80. From p, Fig. 79, set off to u, equal to 
a c. Connect u r, giving the major length to draw Fig. 83, and s u the 
major length for Fig. 81. 

Fig. 81. Let 1 3 equal s u, and form the parallel¬ 
ogram from the tangent s o, all sides being equal. 

Connect 2 4, giving the normal line and minor axis. 

Let 4 5 equal d c as the centre of the mould. Mark 
the width of rail 6 7; through 5 draw a line parallel to 
1 3, and let 5 8 equal the width of the rail ; draw par¬ 
allel to 1 3, cutting the bevel lines for the width of the 
mould at the ends. Find the bevel 4 9 to o, when o 12 
will be the width for the ends. Complete the mould by drawing the 
curve, sections a a and b b showing the bevel applied. 

Fig. 82. To draw the parallelogram. Let 1 2 and 3 4 equal 



Fig. 84. 
The mould. 



a t, Fig. 79, and 1 4 and 2 3 equal l s. Let f w equal the width of 
the rail, cutting the tangent at x. Then x 1. will be the width of the 
















































40 


GENERAL PRACTICE. 


mould at i ; at 3 make it the width of the rail, and complete the mould 
by drawing the curve. For the bevel, take the length of tangent 2 3, 
placing it from 1 to 5; then the angle at 5 will be the bevel for the end 
at 1. The angle at l of the elevation is also the bevel; section c c as 
the bevel is applied, section d d as the rail is taken square through centre 
of the staff. 



Fig. 83. Let 1 3 equal 
u r, Fig. 79, 1 2 and 3 4 
equal o z, and complete the 
parallelogram. The normal 
line is not given on the 
mould, but will find the 
width required on minor 
Fig. 87. -The mould. length. At Fig. 80, let E F 

equal r s, and connect f with the centre; this line will then equal 2 4. 

On this line mark off the points 2 5 7, as they are shown at f x x, 
Fig. 80. For the bevels, from 4 describe the arc 8, equal to d c. 
Draw a line touching at 8, and let 8 9 equal the width of the rail, draw¬ 
ing it parallel to 8, crossing the bevels at 6 o. Find the bevels as indi¬ 
cated from the dotted line 4 to 12 13. The width of the mould at 3 

will be equal to o 13, and at 1 equal to 6 12. Complete the mould by 
drawing the curve. Sections e e and f f show the bevels as applied. 

Fig. 84. From the tangents e b, Fig. 

80, and n r of the elevation, draw the 
mould as before described, finding the 
bevel in like manner. 

Fig. 85. The ramp is drawn as 
shown by transferring the tangent lines with 
a bevel to the board, to be used for the 
pattern. Let v be the end and joint at 
which the mould, Figs. 81 and 83, will 
join, as the straight wood equals v z. Fig. 88.— The mould. 

Always working on the centre line as the tangent, draw the rise line j 
across the face of the pattern as taking in all the winders. The length 
of straight rail may be measured off with the pitchboard. 

A plan of stairs with winders at the landing, having four 
risers in the cylinders, to which straight treads run. In this 
case there will be no ramp, but the mould will have two pitches forming 
a half easement. Scale, ^ inch to the foot. 






















GENERAL PRACTICE. 


4i 


Fig. 86. Let a b c e i be the line of tangents. From d stretch 
out the tangents as shown, making three spaces equal to d c, d h being 
the floor line. From d drop 4 risers contained in the cylinder at k. 
From k draw the elevation of steps and risers outside the cylinder. 
Locate the short 
balusters as shown 
at l m by the dot¬ 
ted line. From this 
line set off half the 
thickness of the rail, 
and draw the tan- 
gen t from these 
points extending to 
n. From the floor 
line at h, set up 5^ 
inches to the centre 
of the rail on the 
landing at j, and 
connect o j, giving 
the length of tan¬ 
gents. For major 
length of mould, 
let s r equal e d 
and connect r t, 
giving length 



re- 


Fig. 87. Let 1 3 


L K J 

Fig. 89. —Starting with one step to a quarter platform. 

quired. For the minor length, proceed in the usual way as shown at 
d e f ; this needs no further explanation. 

equal t r ; 1 2 and 3 4 equal p 0 ; 14 and 3 1 

equal 0 t, and complete the 
parallelogram. For the bevels 
take the length 4 5 and 4 6, 
and place them on the eleva¬ 
tion from w to u y. Lay oft' 
the width of the rail as shown 
below the bevels, and extend 
For the width of the 



Fig. 90.—The mould. 

the line at the side, cutting the bevels at x v, 
mould at the ends, x w for the long tangents at 3, and w v for the short 
tangent at 1 ; make the points on minor length 2 4, the same as from 
d to f. Draw the curve and complete the mould. Let the straight 











































42 


GENERAL PRACTICE. 


wood i 7 equal p z. Sections at 8 and 9 showing the end of the twist 
as squared from the plank. 

Fig. 88. Draw this mould the same as described at Fig. 82. 
Let 1 2 equal a b, and 2 3 equal j t. Find the width for the end from 
the bevel at j, and complete the mould by drawing the curve. Section 
showing the end of the twist as taken from the plank at 4 and 5. 

A plan of stairs starting with one step to a quarter platform in 
the cylinder : no ramp needed for this, as the rail will have two pitches, 
forming a half easement. Scale ^ inch to the foot, 7 T / 2 inch rise, 
10 inch tread, 12 inch cylinder, 2^x3^ inch rail. 

Fig. 8g. From the tangent a b, drop the perpendiculars to h j, 
which will be the floor line. Stretch out the tangents equal to a b c e, 
and at l set up 5^ inches to the centre of the rail at m. From h to n 
set up the three risers contained in the cylinder. At n draw elevation of 
one step, and locate the short balusters n 0. Draw the dotted line for 
bottom of rail; set off half the thickness of rail, draw the tangent extend¬ 
ing to p, and connect m u. From r draw r 
s parallel to k h, let s t equal d e, and connect 
v t, giving major length of mould. For minor 
length, let e g equal r p, and connect g d, giving 
length required. Mark the width required for 
the rail on this line in the usual way. 

Fig. 90. Let 1 3 equal t v, Fig. 89; let 
1 2 and 3 4 equal u v, 1 4 and 3 2 equal u r, 
and complete the parallelogram. On the 
minor length from 2 4 mark the points equal to those on g d. For the 
bevels, take 4 5 and 4 6, and place them from w, as indicated at z y. 
Mark the width of the rail, letting the side cut the bevels on the line 
w y. This gives the width for the end of mould at 7, Fig. 90, and on 
w z is found the width for 3. For the straight wood, from 1, make 1 7 
equal v x, and complete the mould by drawing the curve. Sections 
8 and 9 show the end at joint as squared from the plank. 

Fig. 91. Draw same as Figs. 82 and 88. Let 1 2 equal m r, and 
2 3 equal e f. The bevel and width of mould is found at m. Apply in 
the usual way, draw curve and complete the mould. Sections 4 a»nd 5 
show the end of the twist as squared from the plank. 

A plan of quarter platform stairs having the risers at the 
cylinder placed in such a manner that the rail will run on a continuous 
pitch, requiring no easing. Scale, 1 y 2 inches to the foot. 


















GENERAL PRACTICE. 


43 


Figs. 92 and 94. Plan and elevation. Let a b c be the line 
of tangents, with d as centre and e and f the risers. It will be seen 
that e is outside of the cylinder, while f is inside. The location of e 
must govern the location of f. Let e be located anywhere on the 
tangent line ; then 
to find where f will 
come, take the dis¬ 
tance e b, as here 
shown to be 5 in¬ 
ches ; then the dis¬ 
tance from B to F Fig. 92. 

will be 4 inches, 
making the length 
of one tread from 
e to f. Extend 
a d and b c indefin¬ 
itely, and draw the 
elevation of one rise 
and tread, letting 
a b be the base line. 

From the location 
of the short baluster 
h on plan, produce 
it on the elevation 
at j. From j apply 
the pitch board, as 
shown by the dotted 
line j k, as the bot¬ 
tom line of the rail. 

Set off half the 
thickness of the rail 
and draw the tan¬ 
gent line l m n. 

For the major 

length of the mould, Figs. 92 and 94.— Quarter platform, risers so placed as to give 
from v as centre KAIL ONE PITCH * CENTRE OF rail shown by dotted line. 

describe the curve c o, and connect l 0, giving length required. 

Fig- 93 - Let 1 3 equal l o, and 1 2 and 3 4 equal l m. Produce 
4 in like manner, as all the lines in the parallelogram are one length. 
















































44 


GENERAL PRACTICE. 


For the centre of the curve, let 4 5 equal d g. hind the width of the 
mould at the ends by the parallel lines, as shown, the width at 2 7 being 
3^2 inches. For the bevel, let d p equal 4 6 ; then the angle at p is the 
bevel for both ends. From 1 and 3 add whatever straight wood is 


2 



required—usually 3 to 4 inches is sufficient. Complete the mould by 
drawing the curve in the usual manner. Sections 8 and 9 show the 
bevel applied. 

Fig. 94. To draw the mitre on the rail to intersect a 
turned cap. Let a be the centre of the cap at the intersection of the 
rise and centre line of the rail. Describe the outside circle b c as the 
size of the cap. Draw the arc d e equal to the depth of the turning. 
Let g h be the width of the rail, 3^2 inches ; draw the depth of the mould 
on the rail, as shown at j k, and extend these lines, intersecting the line 

of the cap ; then through the intersec¬ 
tion draw the line of the mitre as m n 
0, and through m n draw the cheek 
line. Draw the elevation of two risers, 
and from the top of the first step set up 
5 inches as the hight ; the cap will rise 
as from l p. Through the point of 
short baluster r apply the pitch board 
for the bottom line of the rail, produce 
the mitre lines on the pattern, t s as 
the cheek and j v as the point of the 
mitre. For the hight of the newel, add 
2 feet 2 inches to p w, giving the hight from the floor to the under side 
of the cap. 

F >g- 95 - Shows an elevation of the pitchboard with the 
easement, having the top curve drawn, also the second rise line extended 
across the pattern, to be used in measuring the length of straight rail. 



























GENERAL PRACTICE. 


45 


A plan of stairs 
commonly called a 
full turn of winders, 

having three pieces in the 
rail to complete the cylin¬ 
der. Scale, Y inch to 
the foot. 

Fig. 96. Plan. Let 

a b c d e f g be the tan¬ 
gents, and h the centre 
from which the curve is 
described. To locate the 
joints c e, set the com¬ 
passes in a, extend to h 
and make the point c. 
Then in g make the point 
e, connect c and e to h ; 
at right angles to c h 
draw the tangent bcd; 
then at e, in the same 
manner, draw the tangent 
d e f. Extend the centre 
line of rail, completing 
the tangent a b and g f. 
Extend g f indefinitely, 
and stretch out the tan¬ 
gents from g to h on the 
line thus made. From f 
as a centre describe e i 
and n j. 

Fig. 97. Elevation. 

Drop perpendiculars from 
g f 1 j indefinitely. From 
any convenient point out¬ 
side of the plan, make the 
hight of the risers on the 
perpendicular g, the six 
risers which is in the 
cylinder as at k l. Then 
at k l draw the elevations 



Figs. 96 and 97.—Full turn of winders three pieces 
IN CYLINDERS. 








































46 


GENERAL PRACTICE. 


of the steps and risers outside of the 
cylinder over which the ramp is 
drawn. Let m n of the elevation be 
the centre of the short balusters, 
also the bottom line of the rail, and 
draw the centre line for the ramp. 
From m in the lower elevation, 
draw a line cutting the centre line 
of the rail at o ; locate a point at p which will be plumb over o, but not 
the short baluster in this case. This is done for the purpose of making 
both ramps the same, requiring only one ramp pattern, and one mould 
for the cylinder, to be used for the three twists, thereby saving time in 
drawing. From o and p as centres, bisect the hight at r ; from r draw 
r s at right angles to k l, and connect s p and s o, giving the lengths of 
tangents required. To find the major length of the mould. From g as 
centre, draw e t, and drop the dotted line indefinitely. From u v draw 
parallels to r s, cutting the dotted line from t at w and x. Connect 
w y and x y, giving the major lengths. Draw the ramp pattern as shown, 
making the joint at z, allowing 3^4 inches from y to be added on the 
mould, Fig. 98. 



Fig. ioi. 

Figs 99 and ioi.—Turnout and quarter platform. 










































GENERAL PRACTICE. 


47 


Fig. 98. I ^et 1 3 equal w k, i 2 
and 3 2 equal u q, and connect 1 2 and 
3 indefinitely, as the tangents. Let 2 4 
equal h f, and connect 4 1 and 4 3, 
giving the chord line. For the centre 
of the mould on the normal, let 4 5 
equal h e, and mark the width of the rail on this line, from the centre 
at 5. Find the width at the ends, in the usual way, and draw the 
straight wood 3 o equal to y z. The point 4 is the centre of the axis. 
Draw the major axis at right angles to 2 4. For the bevel, let 5 7 equal 

6 4 ; the angle at 7 is the bevel for 
all the pieces as applied at the 
sections 8 and 9. Complete the 
mould by drawing the curve with 
a flexible strip or the trammel as 
may be desired. Fig. 98, as it is 
drawn, will be the two pieces to join the ramps for the centre piece on 
the plan, Fig. 96. c d e will be found at the elevation, Fig. 97, as 
u s v for the tangents. Make the joint at 3, cutting off the straight 
wood, which gives the required mould to complete the cylinder. 

A plan of stairs start¬ 
ing with a turnout and hav¬ 
ing a quarter platform. The 
rail to be in two pieces for the 
quarter, forming half easements, 
requiring but one mould which 
will reverse, as the hight will be 
1 y 2 risers in each piece. Scale, inch to the foot. 

Fig* 99- Turnout. Let the face of the first rise be the centre of 
the newel. To find the end of the mould, draw the cap with the mitre 
in the usual way, as the lines indicate at c. From c as the end of the 

mould, draw the dotted line to 
e, at right angles to e c. Draw 
c b, cutting the base line as the 
centre of the rail from a. Then 
abc will be the tangents of 
equal length. Produce d in 
the parallelogram as indicated. 
Place the point of the pitch- 





2 



Fig. 106.—The mould 




















4 8 


.GENERAL PRACTICE. 


board at b and draw the pitch, cutting the perpendicular e a at f. For 
the major length of the mould, from a as centre, describe c g, and con¬ 
nect g f, giving the length required. For the minor length, at right 
angles from b d, set the hight d h equals to a f, and connect h b, giving 
length required. Let j be the centre line of the rail, k one-half the 
width, and erect the perpendiculars, cutting the minor length at l m. 



Fig. ioo. Let i 3 equal g f ; 12 and 3 4 equal f b ; and 1 4 and 
3 2 equal c b. Complete the parallelogram, and connect 2 4. Let 
4562 equal h l m b, then 5 2 will be the width of the mould. Find 
the width at the ends from the bevels, and extend c d indefinitely. Let 
p n equal 4 7, and p o equal 4 8. Draw s r parallel to n o equal to the 
width of the rail. r I hen 0 r will be the width of the mould at 3, and 
n s will be the width for end at 1. Add the straight wood from 1, 




































GENERAL PRACTICE. 


49 


mark the joints at the ends square from the tangents, and complete the 
mould by drawing the curve in the usual manner. 

Fig. ioi. From t, the point from which the curve is described, 
draw the square t a b c, and connect t b, cutting the curve at e. At 
right angles to e b, draw the tangent lines f g, thus making all the tan¬ 
gents of one length. Form the parallelogram at u; from g as centre, des¬ 
cribe e j, and drop the perpendiculars g k and j l. From d set up 



three risers to m. From m and d bisect the hight at n, and from n draw 
parallel to t a, cutting the perpendicular j l at o. Place the point of 
the pitchboard at d, draw the pitch to r, and connect p o, giving the 
tangents for the mould. From m draw mro in the same manner. 
For major length, let s l equal c e, and connect s o, giving length 
required. For minor length, extend o p to t ; let h u equal d t, and 
connect h g, giving the length required. Find the width of the mould 
in the same manner as described in Fig. 99, viz : from bevels. 
































5° 


GENERAL PRACTICE. 


Fig. 102. Let i 3 equal s o ; i 2 and 3 4 equal dp; 1 4 and 3 2 
equal p 0, and connect 2 4. Find the width of the mould 5 2 in the 
same manner as explained in Figs. 99 and 101. To find the bevels, 
let x w equal 4 8 and y w equal 4 7. Parallel to y x, and the width of 
the rail, draw v z, cutting the bevels ; then x z will be the width of the 
mould at 3, and v y for the end at 1. Add the straight w r ood and com¬ 
plete the mould by drawing curve. This mould reverses and completes 
the quarter, 3 being the centre joint over e. 

The amount of the straight may be left on the moulds, so as to join 
them together, or there may be a straight piece put between them. 
This is left to the judgment of the workman. 

A plan of stairs having two obtuse angles, or less than 
a quarter cylinder. Fig. 103 is a platform with risers so placed 
as to have the pitch of the tangents on the same inclination as 

the straight rail. Fig. 105 
has winders at the lower 
portion, with one outside 
of the cylinder and re¬ 
quiring a ramp; the 
upper portion has the 
straight treads running to 
the chord line. Scale, ^ 
inch to the foot. 

Fig. 103. A B C D is 
the portion to draw the 
mould d, the corres¬ 
ponding point to b, forming the parallelogram, and e the centre from 
which the curve is described ; set f b g on the line of the tangents, which 
equals one straight tread. From b as centre, describe c h, giving the 
stretchout of the tangents. From a as centre, describe c j to get the 
major length. At h j drop perpendiculars indefinitely. For the length 
of tangents, place the point of the pitchboard at b, draw to k and extend 
to l ; then k b l is the length. From l draw parallel to h j, cutting the 
perpendicular from j at m ; connect k m, giving the major length. 

Fig. 104. Let 1 3 equal km; 12 and 3 4 equal k b, and 1 4 
and 2 3 equal b l. For the width of the mould in the centre, let 4 5 
equal d n. From 5 as the centre of the mould, mark the width equal 
to the rail, 3^2 inches. For the bevel, let p 0 equal 4 6, and parallel to 
0 f draw s r equal to the width of the rail, cutting the bevel and giving 











GENERAL PRACTICE. 


5* 


the width for the mould at the ends. Add whatever straight wood is 
required, and complete the mould by drawing the curve. 

Fig* 105. From the rise at f drop a perpendicular indefinitely, 
and from any convenient point set the hight required as from g h, six 
risers. Draw the elevation of the two treads outside of the cylinder, 
over which the ramp is to be drawn. From b as centre, describe c j. 
Prom a b j drop perpendiculars indefinitely. At the hight of five risers 



Fig. hi.— Two quarter cylinders, one circular, one elliptical, with elevation. 


draw one straight tread as k l, place the pitchboard with the point at 
the centre of the short baluster as at k. Draw the dotted line for the 
bottom of the rail, set off half the thickness of the rail from k, and draw 
the centre line for the tangent m n, extended to 0. From pr as the centre 
of the short balusters, draw the dotted line for the bottom line of the rail, 
set off half the thickness of the rail, and draw the centre line of the ramp. 
From the short baluster at s, set off half the thickness of the rail, as 





































52 


GENERAL PRACTICE. 


shown at u, and draw n v t, touching at u. Then m n v are the tangents 
for the mould. For major length of the mould, let w x equal a c, and 
connect m x, giving the length required. For the minor length, let d y 





equal o v, and connect y b, giving the length required. From z as the 
centre of the rail, draw the perpendiculars equal to the width of the rail, 
touching the minor length b y, as shown at o o o. 

Fig. 106. Let i 3 equal m x ; 12 and 3 4 equal m n, and 1 4 
and 3 2 equal n v. Connect 2 4 and make the points for the width of 
the mould same as from b y. For the bevels, let 1 q equal 4 5, and let 
100 equal 4 6. Then the angle at q is the bevel for the end at 3, and 
the angle at 1 is the bevel for the end at 1. Find the width at the ends 
in the same manner as already described from bevels. Add the straight 
wood required, and complete the mould by drawing the curve. 

Fig. 107. Draw the ramp from the centre line, as shown ; 
make the length t u so as to join to Fig. 104. 

A plan of stairs having acute angles, or more than a 
quarter cylinder. Fig. 108 is a platform with the risers so placed as 
to have the inclination of the tangents on the same pitch as the straight 
rail. Fig. 109 is the same angle as Fig. 108, but 
having winders with a portion of a tread outside of the 
cylinder, requiring ramps. The mould for the cylinder 
has tangents of equal length, and will be drawn the 
same as for Fig. 108. Scale, inch to the foot. 

Fig. 108. Let abc be the portion to draw the 
mould, with d as the corresponding point to b, forming 
the parallelogram with e as the centre, from which the 
curve is described. Let a j equal one tread, cutting 



Fig. 113. 
Landing mould. 



























GENERAL PRACTICE. 


53 


the plan on the tangent line. Extend c b indefinitely. From b as 
centre, describe j k; make x l equal to one tread. The risers j x will 
have to be bent so as to finish on the string nicely. In this case the 
treads and platform are the same width from a to p. The centres to 
draw the curve of the risers are at r r. For length of the tangents, 
with b as centre, describe c g. From g drop a perpendicular indefinitely. 



From b apply the pitchboard, extending the line to f and m, giving the 
length of the tangents. For the major length of the mould, with a as 
centre, describe c h, and from h drop a perpendicular indefinitely. 
From m draw m n parallel to h g, and connect n f, giving the required 
length. Draw the mould as shown, Fig. iio, using f n for major 
length and f b m for the tangents. This figure will require no further 
explanation. 







































54 


GENERAL PRACTICE. 


Fig. log. Let a l c be the portion to draw the mould, d as the 
corresponding point to l, forming the parallelogram, with e as the 
centre from which the curve is described. From pal drop perpendic¬ 
ulars indefinitely. From any convenient point below p, let k k equal 
five risers as contained in the winders from p s. From k k draw the 

2 elevation of the treads 

outside of the cylinder. 
Locate the centre line 
J— of the rail from kk as 
the centre of the short 
balusters. At any con¬ 
venient point on the 
Fig. 115. —The mould. pitch, as at G G, bisect 

the hight at j. Draw j b parallel to a l, cutting the perpendicular 
from l. At b connect b g; then b g, b m and b f are the tangents for 
the mould. For the major length of mould, from a as centre, describe 
c h. From h drop a perpendicular indefinitely. Draw m n parallel to 
a h, and connect n f, giving the required length. 

Fig. no. Let i 3 equal f n of Fig. 109. Let 1 2 and 3 2 
equal m b and m f; produce 4 in the same 
manner, to complete the parallelogram, as 
all sides are equal. Let 265 equal l o 
e, 5 being the point from which to draw 
the chord line through 1 3. Let 4 9 equal 
d p, draw parallel to 1 3, and mate 4 8 
equal 4 7. The angle at 8 will then be 
the bevel. Find the width of the mould 
at the ends on the bevel in the usual 
manner. 

Draw the ramps from the angles at g, as indicated, t t being the 
joints. 

Draw the mould for Fig. 108 from the major length n f, as shown. 

A plan of stairs having a cylinder constructed of two 
quarters, the landing portion being that of a circle, the low¬ 
er or flight being an ellipse. This is called a “Thumb Ellipse,” 
and is preferable to a true cylinder. It has five winders, but 
requiring no ramp in the rail, as the lower piece will be a 
half easement, forming a graceful, easy turn. Scale ^ inch 
to the foot. 
































GENERAL PRACTICE. 


55 


Fig. hi. Plan. Let a b c e f be the plan of the tangents, with 
g and n as the centres. At the rise r extend a perpendicular indefinitely. 
Let h j equal the hight of six risers as contained in the winders. Draw 
the elevation of straight step and winder outside of the cylinder. Let 
m n be the short baluster, also the bottom line of the rail. Set off half 
the thickness of the rail, and draw the pitch line cutting the perpen¬ 
dicular line c b at o. Let b l on the base line equal b a, and erect a 
perpendicular at l, as indicated by the dotted line, indefinitely. From 
b as centre describe the dotted line, c s, also e p. At s p erect perpendi¬ 
culars indefinitely. Locate the floor line x x from the hight at j on 
the left. From x x set up four inches to the bottom of the rail, with 
half the thickness of the rail, as at r. Draw level tangent r t equal to 
e f, and connect r o, giving tangents required. For the major length of 
mould, from y, draw parallel to step line indefinitely. From the perpendi¬ 
cular l v set off w equal to b k, and connect v w for the length required. 
Extend the tangent line r o to z. For minor length at right angles from 
b k set up k q equal to y z, and connect q b for length required. Find 
the width of mould on minor length in the usual way, as at a 2 b 2 

Fig. 112. Let i 3 equal w v, and i 2 and 3 4 equal 0 v; 14 and 
3 2 equal o y; and connect 2 4. For the bevels let 4 5 equal a b, and 
draw parallel to 1 3, touching the tangents. From 4 describe 6 7, also 
8 9. Then 7 and 9 are the bevels. Let 4 10 11 equal q a2 B2 for the 
width on minor length. Find the width for the ends by the bevels—at 
7 12 for the end at 00 and at 9 13 for the end at 3. Let 2 00 equal 0 u 
for the joint, add the required straightwood from 3, and complete the 
mould by drawing the curve. 

Fig. 113. Let 1 2 and 3 4 equal r t; and 1 4 and 2 3 equal r u. 
For the bevel let 5 6 equal 2 3, the angle at 5 being the bevel. Let the 
end at 3 equal the width of the rail. Find the width of the other end 
by the bevel in the usual way; add the straight wood and complete the 
mould by drawing the curve. 

A plan of stairs starting with four winders, the cylinder 
being a Thumb Ellipse. The flight piece of rail will be an ob¬ 
tuse base. The starting or easement will be acute. The 
straight treads start from the chord line—no ramp required. 
Scale ^ inch to the foot. 

Fig. 114. Let a b e be the portion for the flight or upper piece. 
Draw the parallelogram, all sides equal, to a b. Then a b c d is the 
parallelogram. The starting will be e f g h, with 1 as the centre from 




56 


GENERAL PRACTICE. 



which 
the curve 
is des¬ 
cribed. From b 
as centre draw e 
j c k and f l. 
From l set off to 
m, equal to f g. 
From the points abjkl m 
drop perpendiculars indefi¬ 
nitely. At any convenient 
point below a space off six 
risers, as at n o, draw the ele- 
vation of steps and risers from the per¬ 
pendicular b ; from p and q as the centre 
of the short balusters set off half the thick¬ 
ness of the rail. Draw the centre line of 
the rail as the tangent extended to r. 
From x x as the floor line set up 4 inches 
and half the thickness of the rail at t. Connect t s, 
p giving length of tangents. For the major length of 
Fig. 115, let a 1 z equal a c and connect z b 2, giving 
length required. For the minor length let d c 3 equal 
Y r and connect c 3 b, giving the length required. 
Find the width of the mould on the minor length in the usual way, as 
indicated at d 4 and e 5. For the major length ot Fig. 116 see Fig. 1 14 


FLOOR LINE 


Fig. 


117.—Plan and 

ELEVATION. ALL 
OBTUSE ANGLES. 



























































GENERAL PRACTICE. 


57 



preceding. Let u f 6 equal e g, and connect u v, giving length 
required. For the minor length from h, set up j 7 equal to f 6 v ; 
connect j 7 F f° r the length required. Find the width of the mould 
on minor length in the usual way as at 0 o and p p. 

Fig. 115. Let 1 3 equal z b 2, 1 2 and 3 4 equal s y, 1 4 and 2 3 

equal s b 2, and connect 2 4. Let 456 equal 
c 3, d 4 and e 5. For the bevels draw a line 
parrallel to 1 3, the distance from 4 to be the 
same as from b to the nearest point x x on the 
line a d. Let 4 8 equal 4 7, touching the par¬ 
allel line. The angle at 8 will be the bevel for 
the end at 13. Let 4 12 equal 4 9, touching 

the parrallel line. The angle at 12 will be the 

bevel for the end at 3. For the joint at 13 let 1 
13 equal v y as the borrowed length; mark 
the joint at right angles to the tangent; add 

Fig. 118.— The mould. the straight wood required. Find the width of 
the mould for the ends by the bevels in the usual way, and complete the 
mould by drawing the curve. 

Fig. 116. Let 1 3, equal u v, 1 2 and 3 4 equal t w, 1 4 
and 3 2 equal t v, and connect 2 4. Let 456 equal j 7, 00, pp. 
From 4 draw the parallel line equal to th® 
distance from g to the tangent e f ; in this 
instance it is over the line 1 3 ; or in other 
words, the line 1 3 is the required line. 

Let 4 8 equal 4 7, then the angle at 8 will be 
the bevel for the level end. Let 4 12 equal 
4 9, then the angle at 12 will be the bevel for the end at 3. Add the re¬ 
quired straight wood from 1, find the width of the mould for the ends by 
the bevel in the usual way, and complete the mould by drawing the curve. 

A plan of Elliptical stairs showing the mode of producing 
the tangents on the ground plan from the foci. The starting 
and landing is made the same, having two risers past the 
major axis, but requiring different moulds for the rail, as will 
be seen by the following explanations. Scale ^ inch to the 
foot. The centre line of rail only is drawn in this figure. 

Fig. 117. Plan. Let a b be the major and c d the minor axis. 
The points e f upon the major axis are the foci, and are found thus : 
Take half of the major length, making d e and d f equal to g b. 





















5§ 


GENERAL PRACTICE. 


Locate the joints in the rail at whatever point is desired (in this instance 

a h i b) leaving the other 
two joints on the level to 
be determined hereafter 
from the elevation. At a, 
as the point of the first 
joint occurs on the major 
Fig. iso.—T he mould. axis, draw the tangents at 

right angles to the same indefinitely, and at b in like manner. Draw 
lines from each focus through h indefinitely, and with the compasses 
take any distance on the extended lines, as j and k ; place one foot of 
the compasses in each of these points and describe the arcs bisecting 
at l ; connect l h, which is the joint required. Draw the tangents at 
right angles to the joint; through i draw lines from each focus in the 
same manner. With the compasses set off m n, and from these points 
describe the arcs bisecting at o, and connect o i, giving the joint re¬ 
quired. Draw the tangent at right angles to the joint. Draw the eleva¬ 
tion with the spaces corresponding to 
a p h q i r b. At any convenient 
point locate the second rise, as at s 
on the bottom to the left for a start¬ 
ing point, also locate the seventh 
and twelfth risers as indicated at t u. 

From the centre of the short balus¬ 
ters at s t describe the arcs, through 
which draw the tangents indefinitely. 

At the point of intersection v, which 
is the corresponding point to r of the 
ground plan, Fig. 117, place one foot of the compasses in v, extending to 
w; describe the dotted line w x, connect v x, extending until it touches the 
centre line of the rail on the landing, as at y. From the perpendicular, 
v, set off z equal to r b of the ground plan, Fig. 117. Erect a perpendic¬ 
ular extending to the level tangent line, as at a i. Let b 2 on the ground 
plan equal y a i of the elevation. From 2 draw the tangent touching the 
curve at 3, giving the point at which to make a joint on the level. Pro¬ 
ceed in the same manner as described for hi. For the bottom easement 
extend the tangent downward until it touches the centre line of the rail 
5 y 2 inches above the floor, as at b 2, from which erect a perpendicular. 
Let a d 4 of the ground plan equal b 2 c 3 of the elevation. From n 4 


2 



Fig. 121 —The mould. 


2 























GENERAL PRACTICE. 


59 


draw the tangent touching the curve at e 5, which will be the point at 
which to make the joint. Proceed in the same manner as described for 
h 1 3. Let f 6 b 2 equal d 4 e 5 on the ground plan. Complete the par¬ 
allelograms on the ground plan in the usual way by extending the short 
tangent equal to the long one, as before described in the elliptical pro¬ 
blems .They will be marked without any further explanation in this respect. 



CENTRE OF RAIL 

71 

y 

13 

X 

y 

s 

y 

' FLOOR LINE 



Fig. 129. 

Fig. 118. Bot¬ 
tom mould. 

For the major 
length of the mould let c 3 
g 7 equal a 6 on the 
ground plan. At g 7 erect 
a perpendicular, extending 
to tangent line h 8. Let g 7 1 9 
equal 6 e 5 on the ground plan, 
and connect 1 9 h 8, giving the 
length required. For the minor length, 
let 9 10 on the ground plan equal g 7 h 8 
. of the elevation, and connect 10 d 4, giv¬ 
ing length required. Mark the width of the 
mould on this line in the usual way, as x x 0 0. 
To draw the mould, let 1 3 equal 1 9 h 8, 
Figs. 122, 123, 124, 126, 129 and t 2 and 3 4 equal b 2 h 8, and 1 4 an( i 3 2 

130 . —Circular plan with j , . s 

location of joints equal b 2 and f 6. Connect 2 4, making 4 5° 
and elevation. ^ 


12 




















































6o 


GENERAL PRACTICE. 


equal to io x x o o. For the bevels, from 4, draw the parallel line 
equal to a 9 e 5 on the ground, and 4 8 equal 4 7. The angle at 8 will 
be the bevel for the upper end. Let 4 10 equal 4 9. The angle at 10 
will be the bevel for the wide or level end. Find the width of the mould 
at the end by the bevels in the usual way. To joint the mould at its 
proper point let 2 12 equal b 2 j 9, and complete by drawing the curve. 

Fig. 119. This will also be for the opposite end, and will cover 
the ground plan at 1 r b. From k 0 draw the dotted line at right 
angles to the perpendiculars indefinitely. Let 13 14 equal 13 k o. 
From 14 drop a perpendicular cutting the tangent line at 15. For the 
major length of the mould, let 14 16 equal 5 h on the ground plan, and 
connect 1516, giving length required. Let 1 3 equal 15 16, 1 2 and 

1 4 equal k o 17, 3 2 and 3 4 equal 1517, and connect 2 4. Let 4 5 
equal 12 18, and mark the width of the mould from 18 as the centre. 
For the bevel, from 4 draw the parrallel line equal to the distance be¬ 
tween 12 and the tangent line a p, and let 4 6 equal 4 7. The angle at 
6 will be the bevel for both ends. To joint the mould at its proper 
point let 2 8 equal 17 j 9. Find the width of the mould at the ends in 
the usual way by the bevel. Complete by drawing the curve. 

Fig. 120. Draw this from the minor length. Let 2 4 equal q 19 
of the ground plan. Let 2 1 and 2 3 equal T w, and 41 and 43 equal 

t k 0. From 4 draw the parallel line 5 6 
equal to the distance between 19 and the 
tangent 1 q on the ground plan. Find the 
bevel in the usual manner. Also the width 
for the mould at the ends. Complete the 
mould by drawing the curve. 

Fig. 121. Find the major length for 
the mould at the top of plan, Fig. 117. Let y 20 equal y b 2. From 
20 drop a perpendicular touching the tangent line at 21. Let 20 22 
equal 3 4 on the ground plan, and connect 21 22, giving length re¬ 
quired. For the minor length let 7 8 equal 20 21, and connect 2 8, 
giving length required. Mark on this line the width of the mould in 
the usual manner, as indicated at 23 24. To draw the mould, let 1 3 
equal 21 22, 12 and 3 4 equal 2 3, 1 4 and 3 2 equal y 21 and connect 

2 4. Let 456 equal 8 23 24. From 4 draw the parallel line equal to 
the distance 7, and the tangent line 4 2. For the bevels let 4 8 equal 
4 7 - d he angle at 8 will be the bevel for the level or wide end of the 
mould. Let 4 9 equal 4 10. The angle at 9 will be the bevel for the 


18 



16 


Fig. 125.—The mould. 
Fig. 123—The plan. 
Fig. 126 —The elevation. 










GENERAL PRACTICE. 


6r 


small end. Joint the mould at its proper place, letting 2 12 equal y 25. 
Find the width of the mould for the ends by the bevels in the usual 
manner. Complete by drawing the curve.* 

Circular plan with seventeen risers, giving the centre line 
of rail, requiring five pieces in the rail, and three moulds to be 
drawn. Scale 3 ^ inch to the foot. 

Fig. 122. Plan. With x as centre describe the curve for the face 
of the string, also the dotted line as the centre of the rail. Let l be the 
centre of the newel to describe the cap, draw the mitre, as described in 
Fig. 94. Let a be the point of the mitre, at right angles to x a, draw 
the tangent a b, and draw c b in the same manner. From c to 1 mark 
off for as many spaces as it is desired to have pieces in the rail ; in this 
case we have three, as the joint occurs at c e g i. Through these joints 
draw the tangents. With the square at e x draw e d, then at x c draw 
c d. In like manner draw the other tangents. The location of j and 
the joint k will be found after the elevation is drawn, as in Fig. 117. 

Proceed to draw the elevation thus : Draw the perpendicular at the 
left with ten risers in 
hight, and draw at the 
bottom the horizontal line 0 
as the floor line. Space 
off from the perpendicular 

m m as many divisions as 
there are on the plan, Fig. Fig - I2 7-“ The mould > covering plan from c to i. 

122, from a to f, erecting perpendiculars at each division as indicated. 
Locate the 4th and the icth risers. From the centre of the short balus¬ 
ter, as the bottom line of rail, set off half the thickness of the rail, and 
through these points draw the tangent line 2 6. Draw 1 2 level equal 
to a b, Fig. 123, Plan. From 1 to 6 gives the length and pitch of 
the tangents, equal to a f, Fig. 122. At the perpendicular 4 7 > 
locate the 12th rise, also locate the 16th. From the centre of the 
short baluster set off half the thickness of the rail and draw 
the tangent 6 a to 9 extended. From the floor line set up four 
inches to the bottom of the rail and half the thickness of the rail added, 
as 120. At the intersection at o is the length for the tangents, 1 j, 
on the plan. Draw j k at Fig. 124, Plan, equal to 0 12, completing 
the location and the length of all the tangents. 

* Note.—O n the margin at the left is the hight of 13 risers contained in the 
plan of stairs, with the floor lines and the center line of rail on the levels as shown on 
the elevation. 









62 


GENERAL PRACTICE. 


Fig. 125. To draw the mould, Plan, Fig. 123, Elevation, Fig. 
126, let 0 0 equal a c, and connect o 3, giving the major length 
required. Let 1 3, Fig. 125, equal 30; 12 and 3 4 equal 3 2, 

1 4 and 3 2 equal 1 2, in Fig. 126, and complete the parallelo¬ 
gram. For the width of the mould on the minor length proceed as 
follows: From the point n, Fig. 123, erect a perpendicular n 0 equal to 
0 3, Fig. 126, and connect 0 b, giving the minor length. At p q erect 
perpendiculars, cutting the minor length at r s. Let 4 6 5, Fig. 125, 
equal osr, 57 equal 5 6 ; then 6 7 is the width for the mould. To 
find the bevels, let 4 o equal the distance between n and the tangent a b, 
which, in this instance, is on the major length. Let 4 1 1 equal 4 8 ; 
then the angle at 11 is the bevel for the joint at 1. Let 4 12 equal 
4 9, the angle at 12, which is the bevel for the end at 3. For the 


wi'HtL at the* ends draw 13 14 parrallel to 
ial to p q, and at the cut- 
bevels, giving the width for 
15 and 1 16 equal 11 13, and 
3 equal 12 14. Complete the 
mould by drawing the curve through the 
points thus indicated. 



4 

Fig. 128. — The mould covering 
PLAN, Fig. 124, AND ELEVATION, 
Fig. 129. 


Fig. 127. To draw the Mould. Plan, Fig. 130. This 

mould is drawn from the minor length thus : Draw the horizontal line 
indefinitely, also draw the vertical line. Let 5 6 and 5 7 equal 0 f, 
Fig. 130. From 6 and 7 let all sides of the parallelogram equal the 
tangents 4 5 and 5 6, as shown, 8 9 being the major length, as pro¬ 
duced from the minor length. For the centre and the width of the 
mould, let 5 11 equal 0 p, and from 11 as the centre of the mould 

mark off the width 12 13. For the bevel let 7 15 equal f r, draw 

parallel to 8 9, and let 7 17 equal 7 16 ; the angle at 17 is the bevel 
for both ends. For the width at the ends, from 11 as the centre, 
draw 11 19, parallel to 8 9; then 17 19 is half the width at the ends, 

as 8 o o and 900. Complete the mould by drawing the curve. 

Fig. 128. For the mould. Plan, Fig. 124. Elevation, 
Fig. 129. For the major length let 9 x elevation equal 1 k of Fig. 124. 
From x erect a perpendicular cutting the centre line of rail o 12 at 13, 
and connect 9 13, giving the length required. Find the minor length 
at Fig. 124 by producing the point l in the parallelogram. At l 
drop a perpendicular l m equal to x 13, elevation, and connect m j 
giving the length required. From p as the centre of the rail and o 












GENERAL PRACTICE, 63 


as half the width, drop perpendiculars, cutting the minor length at r s. 
Let 1 3, Fig. 128, equal 9 13, Fig. 129. Let 1 2 and 3 4 equal 0 12, 
complete the parallelogram and connect 2 4 as the minor length equal to 
m j. For the width on minor length, let 4 5 6 equal m rs, and 6 7 equal 
6 5, giving 5 7 as the width required. For the bevels let 4 8 equal the 
distance from l to the tangent j k, and draw parallel to 1 3. Let 4 10 
equal 4 9 ; then the angle at 10 is the bevel for the end at 3. Let 4 12 
equal 4 11; then the angle at 12 is the bevel for the end at 1. For the width 
of the mould at the ends draw 13 14 parallel to 10 12, the distance 
between to equal half the width of the rail as 0 p, Fig. 124. Let loo 
equal 12 14, 3 x x equal 10 13, and complete the mould by drawing 
the curve. 

We now show the principles of Forced Easements, which it is 
necessary to employ under 
certain conditions, where, if 
the mould were drawn in 
the usual way, the proper 
hight would not be re¬ 
tained. Figures 131 to 137 
illustrate the text and make 
the principle plain. Scale, 
y inch to the foot. 

Fig. 131. Ground plan 
for an easement having 
two pitches. The full 
lines in both, this figure 
and the elevation, are 
used. Fig. 133 shows the 
easement as it appears in 
position, as drawn from the 
mould, Fig. 134. Fig.i32will 
accomplish the same result. 

The dotted lines in this fig¬ 
ure are used, as also in the 
elevation. Fig. 135 is the 
mould, and when applied 
and worked will be the 
same as shown at Fig. 133. 

, . , j. Figs. 131, 132 and 133— Ground plan and elevation 

rig. 13b shows the landing POR bottom forced easements. 



w' 

6 











































6 4 


GENERAL PRACTICE. 


of a straight flight, with a quarter cylinder, and the landing rise placed 
near the back of the cylinder. In order to get up to the proper hight 
this figure has been introduced. Fig. 137 shows the mould as drawn. 

Fig. 132. Let e be the centre from which to describe the curve f a 
as the centre line of rail; a b c d, the location of the risers ; connect 
g d, extended, as a base line. At d erect a perpendicular equal to four 
risers, as at 1. At g also erect a perpendicular indefinitely. From the 
centre of the short baluster at 1 find the centre of the rail as at j, and 
apply the pitchboard, cutting the perpendiculars from g at k. From g as 
centre describe a l, and at l erect a perpendicular, cutting the line x x. 
At m connect m k and k n, giving the length and pitch of tangents, to 
draw the mould, Fig. 134. Extend m k to o, and from f as centre 
describe a p. At p erect a perpendicular, cutting the line x x at q, and 
connect n q, giving the major length. Find the minor length, draw the 
parallelogram producing h, connect h g, and at h erect a perpendicular 
h s, equal to 0 n. Connect s g, giving the minor length. Let t be the 
centre of rail, u v the width, and erect perpendiculars cutting the minor 
length at w y z. 

Fig. 134 - Let 1 3 equal n q, i 2 and 3 4 equal k n, 14 and 3 2 

equal m k, and connect 2 4. 
Draw 5 6 parallel to 1 3, and 
let the distance from 4 equal 
the distance from h to the tan¬ 
gent line g f. To find the 
bevels, let 4 6 equal 4 7, when 
Fig. 134.—Mould covering plan, fig. 131. the angle at 6 will be the bevel 
for the end at 1. Let 4 5 equal 4 8, when the angle at 5 will be the 
bevel for the end at 3. For the width of the mould on the minor 
length, let 4 9011 equal s w y z ; then 9 11 is the width. For the 
width at the ends draw 1213 parallel to 6 5 and equal to half the width 
of rail. Let 14 15 and 14 16 equal 6 13, and 3 17 and 3 18 equal 
5 12. Complete the mould by drawing the curve. Fig. 133 will be 
squared on the inside, after which the end will be cut to make the joint 
by applying the bevel at r. This bevel is found by drawing at right 
angles to the pitch m k and the perpendicular m l, giving the bevel as 
at m, and applied at r. 

Fig. 132. Let fi 2 3 be the parallelogram. From f as centre 
describe 2 4, at 4 erect a perpendicular, cutting the line x x at 5, and 
connect 5 n, giving the major length. For the minor length, at 3 erect 
a perpendicular, 3 6 equal to n x, and connect 6 1, giving the length 














GENERAL PRACTICE. 


65 


required. At 1 erect a perpendicular to 7, connect 7 n, giving length 
and pitch of tangent. Find the width on the minor length, as shown 
at 8 9 o. 

Fig. 135- Let a c equal 5 n, a b and c d equal n 7, a d and c b 

equal 1 2, and connect bd. 
Let d e f g equal 6890, 
draw h 1 parallel to a c, 
and from 4 equal to the 
distance 3 and tangent 1 2. 
For the bevel let 1 d equal 
d j ; then the angle at 1 is 
the bevel for the end at a. 
Let r* h equal d k ; the 
angle at h is then the bevel 
for the end at l. Let b l 
equal 1 a, and make joints. 
Find the width at the ends 
in the same manner as for Fig. 134. Complete the mould by drawing 
the curve. 

Fig. 136. From a, centre of the quarter, describe the curve b c. 
Let d c e be the risers. The centre line of rail only is shown here. 
From a and b erect perpendicu¬ 
lars indefinitely. Draw the ele¬ 
vation of steps and risers, and 
from the floor at f set up four 
inches and a half, the thickness 
of the rail, as at g. At d, the 
centre of the short baluster, 
locate the position of the rail. 

From the centre of g draw the 
tangent to any point desired to 
form the ramp, as at h. This is 
left to the judgment of the work¬ 
man. At right angles to g h 
make the joint or cut for the 
twist as 1 j. At right angles to d h mark the joint proper, as k l. 

Fig. 137. For the mould. Let 1 2 equal g m, and 2 3 equal a c. 
Add the straightwood 1 4 equal to m h. Let 5 6 equal j l, as the dis¬ 
tance to mark the joint in the operation of squaring the twist. The 



F'g. 136. —Forced easement over a 

LANDING QUARTER. 



Fig. 135—Mould covering plan fig. 132. 










































66 


SI’LAYEI) WORK. 


5 6 


thickness required for the twist is found at i j. The bevel for squaring 
is at o. In working the twist mark the joint k l ; also mark the thickness 

for the rail and square in from the joint, giv¬ 
ing the direction for the straight rail. Mark 
the top end and apply the mould for the 
curve on the inside. Dress the inside out, 
then take it to a width, after which form the 
easement, squaring the top side first, using 
Fig. i 37 .—The mould. taste in the execution, so as to give it the 

appearance of an ogee. This will be found to give a graceful appear¬ 
ance, and is better than making it in two pieces, as is generally done. 


SPLAYED WORK. 


The seven figures following have been introduced for the 
.purpose of showing how to find the mitres of splayed work, 
such as sills, boxes and hoppers,the mitering of timbers in stairs 
at any angle or pitch, rafters, &c.; also to cut the front string 
of stairs, applying the bevel to cut the mitre from the pitch. 

Fig. 138. A right angle plan with splayed sides, giving the 
side cut, butt and mitre. Let a b be the line of bottom, extended, 
d the angle in plan, b c the incline or splay of sides, c f the thickness of 
the sides, and d m the mitre in plan. From b as centre describe c l. 
From l draw parallel to b d, cutting 
the line of the side e m at r, and 
connect r d. The angle at r is the 
bevel to cut the sides. Draw j g 
parallel to a b, cutting the outside 
edge of side at f. From f draw 
parallel to l r, cutting the line of 
mitre in plan at h. From 11 erect the 
perpendicular h o. Draw g n par¬ 
allel to f h, cutting the perpendicular at o, and connect 0 s. The angle at 
o is the bevel for the mitre. From j draw parallel to b d, cutting the line 
of mitre in plan at k. From k erect the perpendicular, cutting the line g 
o at n, and connect n s. The angle at s is the bevel for the butt joint. 

Fig. 139. Another method, producing the same result 
from plan, the same as that of Fig. 138. Let a b be the line of 

bottom, b c the splay of sides, and c f the thickness of the sides. From 



FlG. 138 —RIGHT ANGLE, GROUND PLAN, 
WITH SPLAYED SILES. 

































SPLAYED WORK. 


67 



FlG. 139. —RIGHT ANGLE, GROUND PLAN, IN 
DIFFERENT FORM. 


b as centre erect the perpendicular and describe c n. From c erect the 
perpendicular c R, draw d r parallel to g c, and connect r b. The angle 
at r is the bevel to cut the sides. Draw j g parallel to b d. From c as 

centre describe g s, cutting the splay 
of side b c, and at s connect s f. The 
>P angle formed at s is the bevel of the 
butt joint. From f as centre describe 
j 0, and connect o s. The angle at o 
is the bevel for the mitre. 

Fig. 140. An obtuse angle in 
plan with splayed sides, giving 
the side cut, butt and mitre for 
the same. Let a b be the line of bottom, and b c the splay of the 
sides, c f the thickness of the sides, and d the angle in plan. Draw 
c m parallel to b d, and e m parallel to d u, being equal to the splay. 
Connect d m, extended, for the mitre 
in plan. From b as centre describe 
c l. From m erect a perpendicular. 

From l draw parallel to c m, cutting 
the perpendicular from m at r, and 
connect r d. The angle at r is the 
bevel for the sides. From f as centre 
describe c g. From h erect a perpen¬ 
dicular. From g draw parallel to f s, 
cutting the perpendicular from h at 0, 
and connect 0 s. The angle at 0 is the bevel for the mitre. The butt 

joint will be found as indicated by the 
dotted lines thus: Extend g f to j, 
h then draw to k, as it cuts the mitre in 
plan. From k draw parallel to the 
side m e, cutting the line f h at p, then 
square over to n, connect n s, and 
the angle at s will be the butt joint as 
required. 

Fig. 141 is an acute plan with 
corresponding points to Fig. 140, 



Fig. 140.—obtuse angle, ground plan, 

WITH SPLAYED SIDES. 


N 



Fig. 141. —acute angle, ground plan, 

WITH SPLAYED SIDES. 


and will need no further explanations. 





































68 


BEVELS AND MITRES. 


BEVELS AND MITRES. 


Fig. 142. To find the mitre or bevel for timbers on the rake 
over a right angle, base or ground plan. Let a b and e f be the 
thickness of the timber to be cut. Let c d be the mitre or angle in 
plan, d l the flight and n a the run. Connect a l, giving the length and 
pitch. Let m n equal g d, and connect l n. The angle at l is the top 
bevel giving the mitre. T he angle at m is the plumb or side cut, and at 
N a is the bevel for the foot. In like 

manner let d o be hight and d f the 
run. Connect 0 f, giving the 
length and pitch. Let 1 ,j equal 1 k, 
connect o j, and giving the top or 
mitre bevel. The angle at 1 is the 


B 



M 



/ 


c 


H '~f\\ 



/ /J 


E F 

Fig. 142.— RIGHT ANGLE GROUND PLAN. 
FIND MITRE FOR TIMBERS ON RAKE. 


TO 


plumb or side bevel. At f is 
the bevel for the foot, and when 
cut will stand plumb over the 
ground plan b g e. 

Fig. 143 is an obtuse angle 
plan: one side only is shown 
in the elevation, as the opposite 
side will be the same. Let a b and k l be the thickness of timber 
to be cut, h g the line of cut in plan. At g erect a perpendicular the 
hight required as g i, and connect 1 l, giving the length and pitch. 
Draw h j parallel to g i, and let j k equal s h. Connect 1 k, giving 
bevel for the top or mitre. The angle at j is the plumb or side bevel, 
and l is the bevel for the 
foot. 

Fig. 147 is an acute 
angle in plan. Let e f and 

r l be the thickness and c d 
the angle or mitre in plan. At 
n erect a perpendicular the 
hight required, as d p. Con¬ 
nect p l, giving the length 
and pitch. From c erect a 
perpendicular parallel to d p. 

Let m n equal c o, and con¬ 
nect p n, giving the top or mitre bevel. At m is the side or plumb 
bevel, and at l the bevel for the foot. 



Fig. 143 —obtuse angle and ground plan. 

Fig. 144.—Acute angle, ground plan, to find 
mitre for timbers on rake. 





































DETERMINING WIDTH OF MOULD IN SPECIAL CASES. 


69 


DETERMINING WIDTH OK MOULD 
IN SPECIAL CASES- 

Figs. 145 and 146, showing how to find the width for the 
mould at the normal point. It is usual to make the mould the 
width of the rail, but in practice it is found necessary to have the mould 
wider, and the question arises, how much will be required ? That will 
depend on the size of the cylinder and the thickness of the rail. For the 
smaller the cylinder and the thicker the rail, the wider will be the mould. 

Fig. 145. Let a be the centre to describe the curve lines, and b g as 
the width of rail in plan. Draw the perpendic¬ 
ular through a and set off on each side half 
the width ofrail, as at d e. At any convenient 
point apply the pitch of the major length. 

Through the point f draw the thickness of 
the rail, and at right angles to the pitch, draw fig. i 45 . | 
the line cutting the thickness of the rail at 
H 1. From the points thus established drop 
perpendiculars cutting the inside of the rail as 
at j l k ; then l m will be the width required 
at the normal point. Let no be the thick¬ 
ness for the plank, and p r will be the width 
for the ends. 

Note.— Draw the mould as explained in the pre¬ 
vious problems, and add whatever is necessary to 
accomplish the desired result in width. After the 
twist is laid out on the plank, reduce the mould to its 
proper size ready to be applied in squaring the twist. 

It will be understood, by referring to j L K, that the 
extra width is on the inside only at the normal line, 
but at the ends it must be on the outside as well. 

Fig. 146 is drawn in the same manner 
as Fig. 145, being the same in plan, width 
and thickness of rail equal (3^ x 3^). 
l m is the width necessary for the mould at 
the normal line, n o the thickness for the 
plank, and p r the width for the ends. It will 

* ’ Figs. 145 and 146.— to find 

be seen that a 2 X A plank will not do to get width of mould when 

u / • r _ _ THICKNESS IS GREATER 

out the rail for Fig. 146, while it is sufficient than width. 

or Fig. 145, but it must be about of an in ch more, or 4^ thick. 





















7 o 


SLIDING THE MOULD. 


SLIDING THE MOULD. 


Fig. 147 shows how to apply the bevels and slide the 

mould. In 
this case the 
bevels are 
alike, the 
tangents be¬ 
ing of one 
length. 

Fig. 148. 
Sliding the 

mould and applying the bevels. This is a quarter with tangents of 
unequal length ; consequently it has two bevels as well. The explana¬ 
tions for these two pieces are the same, both being lettered alike, and 
are laid out for right-hand stairs. 



Fig. 147. —APPLICATION OF BRVELS AND SLIDING OF MOULD FOR QUARTER 
TURN, WITH BOTH PITCH TANGENTS. 


Note.—Dress the top side out of wind, and make the joints to the pattern. 
This is an important feature in rail working, for if it is done carelessly the rail will 
not be as it should, and no matter how well the moulds are drawn the draughtsman 
is sure to be blamed for carelessness or neglect in the working after the rail is hung. 

Operation.—Lay the mould on the piece, keeping the joints even. Let 1234 
be the size of the mould and twist at 9 square through, as 9 9. With the stock of the 
bevel in the left hand, the blade to the right and the concave side of the twist also to 

the right, apply the bevel 
at a through the centre. 
Now turn the twist, let¬ 
ting concave side be 
to the left, but keep 
the bevel in the same 
position, applying it at 
B. In the same manner 
then mark the square of 
the rail as shown. Now 
at o, where the bevel 
touches the face, square 
over on the twist as the tangent, and show by the dotted lines. Then slide the 
mould along the tangent line as shown, until it covers the same points on the 
opposite end. Now the mould, as it is lying on the twist, is shown by 5 6 and 7 8. 
Mark the inside and outside along the edge of the pattern as shown by the dotted 
lines, and the overwood is shown by the shaded portion along the face and at the 
end, sections D and C. Now turn the twist over and repeat the operation from the 



















































































SLIDING TIIE MOULD. 


7 i 


opposite end, and x x will be the section to cut off. Now put the piece in the 
vise and with the ax or gouge and mallet remove the surplus wood, then with a plane 
dress out the inside, finishing up with a spoke shave. The outside will be done in the 
same manner. Now, having the twist to a width, mark the top line, using a strip 
ot paste board, letting it lie close to the wood. 

Note.—Fig. 147 has straightwood on both ends and Fig. 148 on one end, and 
must be squared in from the joint equal to the amount of straightwood intersecting 
with the line of the paste board, and forming an angle which must be eased. This 
is done according to the taste of the workman, and blending with the piece on 
which it will be jointed. A little practice will soon make the workman understand 
what is needed to make a good and graceful rail. Short and quick easements and 
too much straightwood are to be avoided, as they cripple the rail to the eye. 

Now having the line for the top marked on the inside and outside, remove the 
overwood and finish off with a spoke shave ; after which guage to thickness, using 
your fingers as calipers through the centre, as it is quite certain to be thick in the 
centre, and when moulded giving a clumsy appearance, which is to be avoided. 

Fig. 149 is a starting easement or turnout piece. This is 
laid out for the left hand and the overwood is marked on the 
back edge. Both bevels are applied from the same side, and the stock 
to the back or outside of the piece, a b c d is the twist, and the mould 
5 5 is the cen¬ 
tre line of the 
stuff and tan¬ 
gent. e f is 
the section of 
the size of rail, 

0 o the tangent 
as marked on Fig. 149.— turnout, 

the piece from the bevel as it comes to the face, and is shown by the 
dotted lines. This mould will slide on the bottom joint as a b, giving 
its new position as shown by 1 2 and 3 4. On the upper end 5 o is the 
distance the mould has moved. Repeat the operation on the other 
side, and j l will be the inside portion to remove. After the inside 
is dressed out the easement will be formed by its own lines, as they occur 
along its face, and will need no further explanation. 













































72 


THE NONPAREIL SYSTEM IN ITS BRIEFEST FORM. 


the: nonpareil system in its 

BRIEFEST FORM. 

Fig. 150. A quarter circle ground plan, with tangents in 

elevation of one pitch. Let a b c d 

be the ground of tangents, and a b f 
the stretchout of the tangents on the 
base line. Let a i be the hight and 
1 2 f the tangents in elevation. From 
a as centre and a c as a radius, pro¬ 
duce e, and connect e i, giving the 
major length. Now find the position 
of the tangents for the mould. From 

1 as centre and 1 e as a radius, de¬ 
scribe the curve indefinitely; then from 

2 as a centre and 2 Fas a radius, de¬ 
scribe the curve touching at 3. Con¬ 
nect 2 3, giving the required form for 
the mould. Produce the point 4 in 
the usual way. To find the bevel, set 
the compasses the length of the dotted 
line o o, set one foot in 4 and describe 
the segment. Then square from the 

tangent, touching the point 4. Now, 
from the intersection with the tan¬ 
gent draw a line touching the seg¬ 
ment, and the angle thus formed is 
the bevel to be applied at both ends 
of the twist. 

Fig. 151 is drawn from the 
same base line as that of Fig. 

150, this having tangents of 
different pitches. Find all points 
in this in the same manner as for 
Fig. 150, all letters and numbers 
being the same. The two bevels will 
be found from the two tangents by 
drawing a line touching the segment. 

Fig. 152. A segment of a 



PI.AN OF FIG. 150, WITH TANGENTS 
AND BEVELS FOR SAME. 



Fig. 150 —RIGHT ANGLE, GROUND PLAN, 
ELEVATION TANGENTS OF MOULD 
AND BEVEL. 



















THE NONPAREIL SYSTEM IN ITS BRIEFEST FORM. 


73 


circle with tangents of different pitches in elevation. Let a b 


c d be the ground plan,, a b f the 



stretchout of the tangents on the 



Fig. 154. —ELEVATION, BEVEL AND TAN 
GENTS OF ONE PITCH, TO COVER 
GROUND PLAN OF 
FIG. 152. 


base line, a i the hight, and 1 2 f the tangents in elevation. From a as 
centre and a c as a radius, produce e. Connect e i, giving the major 
length. Now find the position of the tangents for the mould. From 



1 as a centre and 1 e as a radius describe the curve indefinitely ; 
then from 2 as centre and 2 f as a radius describe the curve touching 





























74 


THE NONPAREIL SYSTEM IN ITS BRIEFEST FORM. 


at 3, and connect 2 3, giving the required form for the mould. Find 
the point 4 in the usual manner. For the bevels set the compasses the 
length of the dotted line o 0, set one foot in 4 and describe the segment. 
Find the bevel by squaring from the tangents, touching at 4. From the 
intersection with the tangents draw a line touching the segment, and 
the angles thus formed give the bevels. 

Fig. 153 is drawn from the same base line as Fig. 152, 
having one pitch and one bevel tangent. Find all points and 



J -v-V- \X _LsZJ_ _ 

F EB A F EB A 


Fig. 156.—ELEVATION, BEVEL, RAKE AND 
LEVEL TANGENTS, TO COVER 
GROUND PLAN OF 
FIG. 155. 


FlG. 157.—ELEVATION, BEVEL, TANGENTS 
OF ONE PITCH, TO COVER 
GROUND PLAN OK 
FIG. 155. 


positions in the same manner as in Fig. 152. 1 3 is the length of 

the mould and 1 b 3 the position of the tangents in the mould. 

Fig. 154. This also is drawn from the same base line, 

having its tangents of the same pitch, requiring but one bevel, and 
will be found in the same manner as Figs. 152 and 153. 

Figs. 155, 156 and 157. These figures are drawn in the same 
manner as those already described, and will all stand over the 
ground plan (Fig. 155). All descriptive letter press being the same, it 
will need no further explanation, as the system is universal— Non¬ 
pareil in Minimum. 




















INDEX. 


GEOMETRICAL PROBLEMS. 

PAGE 

Triangles. i 

Perpendiculars. 2 

Stretch-out of cylinder. 2 

Bisecting a line. 2 

Finding the radius to describe a circumference through any three 

points not in a straight line. 3 

Circumference, diameter, radius, chord, segment and tangent. 3 

Drawing the ellipse with a string. 4 

Tangent to a circle. 4 

Tangent to an ellipse. 4 

Spirals.5 & 6 

Ellipse drawn with a trammel and rod. 6 

Finding radius. 7 

Finding the octagon of any square. 7 

To draw an octagon on a given side. 8 

HAND-RAILING. 

Length of mould. 8 

Stretch-out of tangents. 8 

Forming the mould. 8 

Drawing the curve line by the use of trammel. 8 

Three elevations from one plan—namely, one pitch and one level; 

one with equal pitches and one with unequal pitches. 9 

Finding the position of the axis. . 11 

Finding the centre of curve on the major axis. n 

Finding the length of trammel rod. 11 

ILLUSTRATIVE PROBLEMS. 

Quarter circle with equal pitches. n 

Quarter circle with unequal pitches. 13 

Quarter circle with one pitch and one level tangent. 15 




























INDEX. 


77 


Obtuse angle base with equal pitches. 

“ “ “ “ unequal “ .. 

one pitch and one level tangent. 

Acute angle base with equal pitches. 

“ “ “ “ unequal “ . 

one pitch and one level tangent. 

Right angle elliptical base with tangents of equal pitch. 

“ “ “ “ “ “ “ unequal “. 

“ “ single pitch; level tangent. 

Obtuse angle elliptical base with tangents of equal pitch. 

“ “ “ “ “ unequal tangents. 

“ “ “ “ single pitch and level tangent... 

Acute angle elliptical base with tangents of equal pitch. 

“ “ “ “ “ •“ “ unequal pitch. 

“ “ “ “ single pitch and level 


PAGE 
16 

17 

18 

1 9 

20 

2 [ 
22 

* J 

24 

25 

26 

27 
2S 

29 

30 


GENERAL PRACTICE. 


Plan and elevation of string and position of rail on a straight flight 

of stairs. 

Starting easement from a cap, obtuse. 

“ “ “ “ “ acute. 

Landing with one rise in cylinder.. 

“ “ winders and a ramp in the rail.. 

“ “ “ without lamp in the rail. 

Platform at bottom with two risers in the cylinder and without 

ramp in the rail. 

Quarter turn platform with newel cap and easement. . 

Full turn of winders with three pieces in the cylinder, with ramps 

in the rail. 

Starting easement and platform with rail of two pitches, obtuse 

angle. 

Obtuse angle platform; also obtuse angle winders with ramp in the 

rail. 

Acute angle with winders ; also acute angle platform.. 

“ Thumb ” elliptic; right angle base with two pitches and without 

ramp in the rail. 

‘‘Thumb” elliptic; acute and obtuse angle bases with winders and 
two pitches, without ramp in the rail. 


3 1 

33 

34 

35 
37 
39 

41 

43 

45 

46 

48 

49 

5i 

53 
































73 


INDEX. 


PAGE 

Elliptical stairs; plan and elevation, all obtuse angles.55 & 56 

Circular stairs with location of joints and elevation.59 & 61 

Forced easement at bottom. 63 

“ “ quarter landing. 65 

SPLAYED WORK. 

Right angle ground plan with splayed sides. 66 

Obtuse “ “ “ “ “ “ . 67 

Acute “ “ “ “ “ “ . 67 


BEVELS AND MITRES. 

To find the mitre for timber on rake with right angle ground plan. 68 

“ “ “ “ “ “ “ “ “ obtuse “ ‘ £ “ . 68 

“ “ “ “ “ “ “ “ “ acute “ “ “ . 68 

DETERMINING WIDTH OF MOULD IN 

SPECIAL CASES. 

To find width of mould when thickness is greater than width. 69 

SLIDING THE MOULD. 

Application of bevels and sliding of mould for quarter turn with 


tangents of equal pitch. 70 

Landing over an obtuse angle. 70 

Turn-out. 71 


THE NONPAREIL SYSTEM IN ITS BRIEFEST 

FORM. 

Right angle ground plan, elevation tangents of mould and bevel. 72 

Obtuse plan, bevels and tangents of different pitches. 73 

Acute “ “ “ “ “ “ “ . 72 
























"N 




















